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Quadratic reciprocity
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==Motivating examples== Quadratic reciprocity arises from certain subtle factorization patterns involving perfect square numbers. In this section, we give examples which lead to the general case. ===Factoring ''n''<sup>2</sup> β 5=== Consider the polynomial <math>f(n) = n^2 - 5</math> and its values for <math>n \in \N.</math> The prime factorizations of these values are given as follows: {| class="wikitable" style="text-align:right;" |- ! ''n'' !! colspan=2 | {{tmath|f(n)}} || !! ''n'' !! colspan=2 | {{tmath|f(n)}} || !! ''n'' !! colspan=2 | {{tmath|f(n)}} |- | 1 || β4 || β2<sup>2</sup> || || 16 || 251 || 251 || || 31 || 956 || 2<sup>2</sup>⋅239 |- | 2 || β1 || β1 || || 17 || 284 || 2<sup>2</sup>⋅71 || || 32 || 1019 || 1019 |- | 3 || 4 || 2<sup>2</sup> || || 18 || 319 || 11⋅29 || || 33 || 1084 || 2<sup>2</sup>⋅271 |- | 4 || 11 || 11 || || 19 || 356 || 2<sup>2</sup>⋅89 || || 34 || 1151 || 1151 |- | 5 || 20 || 2<sup>2</sup>⋅5 || || 20 || 395 || 5⋅79 || || 35 || 1220 || 2<sup>2</sup>⋅5⋅61 |- | 6 || 31 || 31 || || 21 || 436 || 2<sup>2</sup>⋅109 || || 36 || 1291 || 1291 |- | 7 || 44 || 2<sup>2</sup>⋅11 || || 22 || 479 || 479 || || 37 || 1364 || 2<sup>2</sup>⋅11⋅31 |- | 8 || 59 || 59 || || 23 || 524 || 2<sup>2</sup>⋅131 || || 38 || 1439 || 1439 |- | 9 || 76 || 2<sup>2</sup>⋅19 || || 24 || 571 || 571 || || 39 || 1516 || 2<sup>2</sup>⋅379 |- | 10 || 95 || 5⋅19 || || 25 || 620 || 2<sup>2</sup>⋅5⋅31 || || 40 || 1595 || 5⋅11⋅29 |- | 11 || 116 || 2<sup>2</sup>⋅29 || || 26 || 671 || 11⋅61 || || 41 || 1676 || 2<sup>2</sup>⋅419 |- | 12 || 139 || 139 || || 27 || 724 || 2<sup>2</sup>⋅181 || || 42 || 1759 || 1759 |- | 13 || 164 || 2<sup>2</sup>⋅41 || || 28 || 779 || 19⋅41 || || 43 || 1844 || 2<sup>2</sup>⋅461 |- | 14 || 191 || 191 || || 29 || 836 || 2<sup>2</sup>⋅11⋅19 || || 44 || 1931 || 1931 |- | 15 || 220 || 2<sup>2</sup>⋅5⋅11 || || 30 || 895 || 5⋅179 || || 45 || 2020 || 2<sup>2</sup>⋅5⋅101 |} The prime factors <math>p</math> dividing <math>f(n)</math> are <math>p=2,5</math>, and every prime whose final digit is <math>1</math> or <math>9</math>; no primes ending in <math>3</math> or <math>7</math> ever appear. Now, <math>p</math> is a prime factor of some <math>n^2-5</math> whenever <math>n^2 - 5 \equiv 0 \bmod p</math>, i.e. whenever <math>n^2 \equiv 5 \bmod p,</math> i.e. whenever 5 is a quadratic residue modulo <math>p</math>. This happens for <math>p= 2, 5</math> and those primes with <math>p\equiv 1, 4 \bmod 5,</math> and the latter numbers <math>1=(\pm1)^2</math> and <math>4=(\pm2)^2</math> are precisely the quadratic residues modulo <math>5</math>. Therefore, except for <math>p = 2,5</math>, we have that <math>5</math> is a quadratic residue modulo <math>p</math> iff <math>p</math> is a quadratic residue modulo <math>5</math>. The law of quadratic reciprocity gives a similar characterization of prime divisors of <math>f(n)=n^2 - q</math> for any prime ''q'', which leads to a characterization for any integer <math>q</math>. ===Patterns among quadratic residues=== Let ''p'' be an odd prime. A number modulo ''p'' is a [[quadratic residue]] whenever it is congruent to a square (mod ''p''); otherwise it is a quadratic non-residue. ("Quadratic" can be dropped if it is clear from the context.) Here we exclude zero as a special case. Then as a consequence of the fact that the multiplicative group of a [[finite field]] of order ''p'' is cyclic of order ''p-1'', the following statements hold: *There are an equal number of quadratic residues and non-residues; and *The product of two quadratic residues is a residue, the product of a residue and a non-residue is a non-residue, and the product of two non-residues is a residue. For the avoidance of doubt, these statements do ''not'' hold if the modulus is not prime. For example, there are only 3 quadratic residues (1, 4 and 9) in the multiplicative group modulo 15. Moreover, although 7 and 8 are quadratic non-residues, their product 7x8 = 11 is also a quadratic non-residue, in contrast to the prime case. Quadratic residues appear as entries in the following table, indexed by the row number as modulus and column number as root: {| class="wikitable" style="text-align:right;" cellpadding="2" |+ Squares mod primes |- ! ''n'' | 1||2|| 3|| 4||5|| 6|| 7|| 8 ||9 ||10|| 11|| 12|| 13|| 14 ||15|| 16|| 17|| 18 ||19|| 20|| 21|| 22|| 23|| 24 ||25 |- ! ''n''<sup>2</sup> |1 |4 |9 |16 |25 |36 |49 |64 |81 |100|| 121|| 144|| 169|| 196 ||225|| 256|| 289|| 324 ||361 || 400|| 441|| 484|| 529|| 576 ||625 |- ! mod 3 | 1 || 1 || 0 | 1 || 1 || 0 | 1 || 1 || 0 | 1 || 1 || 0 | 1 || 1 || 0 | 1 || 1 || 0 | 1 || 1 || 0 | 1 || 1 || 0 | 1 |- ! mod 5 | 1 || 4 || 4 || 1 || 0 | 1 || 4 || 4 || 1 || 0 | 1 || 4 || 4 || 1 || 0 | 1 || 4 || 4 || 1 || 0 | 1 || 4 || 4 || 1 || 0 |- ! mod 7 | 1 || 4 || 2 || 2 || 4 || 1 || 0 | 1 || 4 || 2 || 2 || 4 || 1 || 0 | 1 || 4 || 2 || 2 || 4 || 1 || 0 | 1 || 4 || 2 || 2 |- ! mod 11 | 1 || 4 || 9 || 5 || 3 || 3 || 5 || 9 || 4 || 1 || 0 | 1 || 4 || 9 || 5 || 3 || 3 || 5 || 9 || 4 || 1 || 0 | 1 || 4 || 9 |- ! mod 13 | 1 || 4 || 9 || 3 || 12 || 10 || 10 || 12 || 3 || 9 || 4 || 1 || 0 | 1 || 4 || 9 || 3 || 12 || 10 || 10 || 12 || 3 || 9 || 4 || 1 |- ! mod 17 | 1 || 4 || 9 || 16 || 8 || 2 || 15 || 13 || 13 || 15 || 2 || 8 || 16 || 9 || 4 || 1 || 0 | 1 || 4 || 9 || 16 || 8 || 2 || 15 || 13 |- ! mod 19 | 1 || 4 || 9 || 16 || 6 || 17 || 11 || 7 || 5 || 5 || 7 || 11 || 17 || 6 || 16 || 9 || 4 || 1 || 0 | 1 || 4 || 9 || 16 || 6 || 17 |- ! mod 23 | 1 || 4 || 9 || 16 || 2 || 13 || 3 || 18 || 12 || 8 || 6 || 6 || 8 || 12 ||18 || 3 || 13 || 2 || 16 || 9 || 4 || 1 || 0 | 1 || 4 |- ! mod 29 | 1 ||4|| 9|| 16|| 25|| 7|| 20|| 6 ||23 ||13|| 5|| 28|| 24|| 22 ||22|| 24|| 28|| 5 ||13 || 23|| 6|| 20|| 7|| 25 ||16 |- ! mod 31 | 1 ||4|| 9|| 16||25|| 5|| 18|| 2 ||19 ||7|| 28|| 20|| 14|| 10 ||8|| 8|| 10|| 14 ||20 || 28|| 7|| 19|| 2|| 18 ||5 |- ! mod 37 | 1 ||4|| 9|| 16||25|| 36|| 12|| 27 ||7 ||26||10|| 33|| 21|| 11 ||3|| 34|| 30|| 28 ||28 || 30|| 34|| 3|| 11|| 21 ||33 |- ! mod 41 | 1 ||4|| 9|| 16||25|| 36|| 8|| 23 ||40 ||18||39||21|| 5|| 32 ||20|| 10|| 2|| 37 ||33 || 31|| 31|| 33|| 37|| 2 ||10 |- ! mod 43 | 1 ||4|| 9|| 16||25|| 36|| 6|| 21 ||38 ||14|| 35|| 15|| 40|| 24 ||10|| 41|| 31|| 23 ||17 || 13|| 11 || 11|| 13|| 17 ||23 |- ! mod 47 | 1 ||4|| 9|| 16||25|| 36|| 2|| 17 ||34 ||6|| 27|| 3|| 28|| 8 ||37||21||7||42||32||24||18|| 14|| 12 || 12 ||14 |} This table is complete for odd primes less than 50. To check whether a number ''m'' is a quadratic residue mod one of these primes ''p'', find ''a'' β‘ ''m'' (mod ''p'') and 0 β€ ''a'' < ''p''. If ''a'' is in row ''p'', then ''m'' is a residue (mod ''p''); if ''a'' is not in row ''p'' of the table, then ''m'' is a nonresidue (mod ''p''). The quadratic reciprocity law is the statement that certain patterns found in the table are true in general. ===Legendre's version=== Another way to organize the data is to see which primes are quadratic residues mod which other primes, as illustrated in the following table. The entry in row ''p'' column ''q'' is '''R''' if ''q'' is a quadratic residue (mod ''p''); if it is a nonresidue the entry is '''N'''. If the row, or the column, or both, are β‘ 1 (mod 4) the entry is blue or green; if both row and column are β‘ 3 (mod 4), it is yellow or orange. The blue and green entries are symmetric around the diagonal: The entry for row ''p'', column ''q'' is '''R''' (resp '''N''') if and only if the entry at row ''q'', column ''p'', is '''R''' (resp '''N'''). The yellow and orange ones, on the other hand, are antisymmetric: The entry for row ''p'', column ''q'' is '''R''' (resp '''N''') if and only if the entry at row ''q'', column ''p'', is '''N''' (resp '''R'''). The reciprocity law states that these patterns hold for all ''p'' and ''q''. {| class="wikitable" style="text-align:center; border:3px solid black;" |+ Legend |- |width="25" style="background:#98ff98;" |'''R''' | align="left" | ''q'' is a residue (mod ''p'') |rowspan="2" | ''q'' β‘ 1 (mod 4) or ''p'' β‘ 1 (mod 4) (or both) <br> (reciprocating) |- |width="25" style="background:#ADD8E6;" |'''N''' || align="left" | ''q'' is a nonresidue (mod ''p'') |- |width="25" style="background:#FFCC99;" |'''R''' | align="left" | ''q'' is a residue (mod ''p'') | rowspan="2" | ''q'' β‘ ''p'' β‘ 3 (mod 4) <br> (non-reciprocating) |- |width="25" style="background:#FBEC5D;" |'''N''' || align="left" | ''q'' is a nonresidue (mod ''p'') |} {| class="wikitable" style="text-align:center; border:3px solid black;" |- | colspan="2" rowspan="2" | ! colspan="24" | ''q'' |- !width="25" | 3 !width="25" | 5 !width="25" | 7 !width="25" |11 !width="25" |13 !width="25" |17 !width="25" |19 !width="25" |23 !width="25" |29 !width="25" |31 !width="25" |37 !width="25" |41 !width="25" |43 !width="25" |47 !width="25" |53 !width="25" |59 !width="25" |61 !width="25" |67 !width="25" |71 !width="25" |73 !width="25" |79 !width="25" |83 !width="25" |89 !width="25" |97 |- !rowspan="24" width="25" | ''p'' !width="25" | 3 ! <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#98ff98;" |'''R'''<!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#FFCC99;" |'''R'''<!-- 19 --> |style="background:#FBEC5D;" |'''N'''<!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#FFCC99;" |'''R'''<!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N'''<!-- 41 --> |style="background:#FFCC99;" |'''R'''<!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N'''<!-- 53 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FBEC5D;" |'''N''' <!-- 83 --> |style="background:#ADD8E6;" |'''N'''<!-- 89 --> |style="background:#98ff98;" |'''R'''<!-- 97 --> |- ! 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<!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#98ff98;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !43 |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 11 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#FFCC99;" |'''R''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> ! <!-- 43 --> |style="background:#FFCC99;" |'''R''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !47 |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> ! <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !53 |style="background:#ADD8E6;" |'''N''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#98ff98;" |'''R''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> ! <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !59 |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> ! <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FBEC5D;" |'''N''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !61 |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> ! <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#98ff98;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !67 |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FFCC99;" |'''R''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> ! <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#FBEC5D;" |'''N''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !71 |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#FFCC99;" |'''R''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> ! <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !73 |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 19 --> |style="background:#98ff98;" |'''R''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 67 --> |style="background:#98ff98;" |'''R''' <!-- 71 --> ! <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !79 |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 11 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#FFCC99;" |'''R''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> ! <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !83 |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#FFCC99;" |'''R''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#FBEC5D;" |'''N''' <!-- 79 --> ! <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !89 |style="background:#ADD8E6;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 67 --> |style="background:#98ff98;" |'''R''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 83 --> ! <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !97 |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> ! <!-- 97 --> |} Ordering the rows and columns mod 4 makes the pattern clearer. {| class="wikitable" style="text-align:center; border:3px solid black;" |- | colspan="2" rowspan="2" | ! colspan="24" | ''q'' |- !width="25" |83 !width="25" |79 !width="25" |71 !width="25" |67 !width="25" |59 !width="25" |47 !width="25" |43 !width="25" |31 !width="25" |23 !width="25" |19 !width="25" |11 !width="25" | 7 !width="25" | 3 !width="25" | 5 !width="25" |13 !width="25" |17 !width="25" |29 !width="25" |37 !width="25" |41 !width="25" |53 !width="25" |61 !width="25" |73 !width="25" |89 !width="25" |97 |- !rowspan="24" width="25" | ''p'' !83 ! <!-- 83 --> |style="background:#FBEC5D;" |'''N''' <!-- 79 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FFCC99;" |'''R''' <!-- 31 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 11 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !79 |style="background:#FFCC99;" |'''R''' <!-- 83 --> ! <!-- 79 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FFCC99;" |'''R''' <!-- 31 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 11 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !71 |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> ! <!-- 71 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#FFCC99;" |'''R''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#FBEC5D;" |'''N''' <!-- 23 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !67 |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#FBEC5D;" |'''N''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> ! <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#FFCC99;" |'''R''' <!-- 47 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !59 |style="background:#FBEC5D;" |'''N''' <!-- 83 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> ! <!-- 59 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#FBEC5D;" |'''N''' <!-- 23 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !47 |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> ! <!-- 47 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#FBEC5D;" |'''N''' <!-- 23 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !43 |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#FFCC99;" |'''R''' <!-- 47 --> ! 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<!-- 31 --> |style="background:#FBEC5D;" |'''N''' <!-- 23 --> |style="background:#FFCC99;" |'''R''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> !style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !23 |style="background:#FBEC5D;" |'''N''' <!-- 83 --> |style="background:#FBEC5D;" |'''N''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#FFCC99;" |'''R''' <!-- 47 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FFCC99;" |'''R''' <!-- 31 --> ! <!-- 23 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !19 |style="background:#FFCC99;" |'''R''' <!-- 83 --> |style="background:#FBEC5D;" |'''N''' <!-- 79 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#FBEC5D;" |'''N''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#FFCC99;" |'''R''' <!-- 47 --> |style="background:#FFCC99;" |'''R''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> ! <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 11 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- ! 11 |style="background:#FBEC5D;" |'''N''' <!-- 83 --> |style="background:#FBEC5D;" |'''N''' <!-- 79 --> |style="background:#FFCC99;" |'''R''' <!-- 71 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FFCC99;" |'''R''' <!-- 59 --> |style="background:#FFCC99;" |'''R''' <!-- 47 --> |style="background:#FBEC5D;" |'''N''' <!-- 43 --> |style="background:#FFCC99;" |'''R''' <!-- 31 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> ! <!-- 11 --> |style="background:#FBEC5D;" |'''N''' <!-- 7 --> |style="background:#FFCC99;" |'''R''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !7 |style="background:#FBEC5D;" |'''N''' <!-- 83 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FFCC99;" |'''R'''<!-- 71 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#FFCC99;" |'''R''' <!-- 43 --> |style="background:#FBEC5D;" |'''N''' <!-- 31 --> |style="background:#FFCC99;" |'''R''' <!-- 23 --> |style="background:#FBEC5D;" |'''N''' <!-- 19 --> |style="background:#FFCC99;" |'''R''' <!-- 11 --> ! <!-- 7 --> |style="background:#FBEC5D;" |'''N''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !3 |style="background:#FBEC5D;" |'''N''' <!-- 83 --> |style="background:#FFCC99;" |'''R''' <!-- 79 --> |style="background:#FBEC5D;" |'''N''' <!-- 71 --> |style="background:#FFCC99;" |'''R''' <!-- 67 --> |style="background:#FBEC5D;" |'''N''' <!-- 59 --> |style="background:#FBEC5D;" |'''N''' <!-- 47 --> |style="background:#FFCC99;" |'''R'''<!-- 43 --> |style="background:#FFCC99;" |'''R'''<!-- 31 --> |style="background:#FBEC5D;" |'''N'''<!-- 23 --> |style="background:#FFCC99;" |'''R'''<!-- 19 --> |style="background:#FBEC5D;" |'''N''' <!-- 11 --> |style="background:#FFCC99;" |'''R''' <!-- 7 --> ! <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#98ff98;" |'''R'''<!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N'''<!-- 41 --> |style="background:#ADD8E6;" |'''N'''<!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N'''<!-- 89 --> |style="background:#98ff98;" |'''R'''<!-- 97 --> |- ! 5 |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#98ff98;" |'''R''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#98ff98;" |'''R''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#98ff98;" |'''R'''<!-- 31 --> |style="background:#ADD8E6;" |'''N'''<!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 19 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 3 --> ! <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#98ff98;" |'''R'''<!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R'''<!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- ! 13 |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> ! <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !17 |style="background:#98ff98;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 67 --> |style="background:#98ff98;" |'''R''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#98ff98;" |'''R'''<!-- 13 --> ! <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !29 |style="background:#98ff98;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#98ff98;" |'''R''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 67 --> |style="background:#98ff98;" |'''R''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#98ff98;" |'''R''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> ! <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !37 |style="background:#98ff98;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#98ff98;" |'''R''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#98ff98;" |'''R''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> ! <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !41 |style="background:#98ff98;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#98ff98;" |'''R''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#98ff98;" |'''R''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> ! <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#ADD8E6;" |'''N''' <!-- 97 --> |- !53 |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#98ff98;" |'''R''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#98ff98;" |'''R''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#98ff98;" |'''R''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> ! <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#ADD8E6;" |'''N''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !61 |style="background:#98ff98;" |'''R''' <!-- 83 --> |style="background:#ADD8E6;" |'''N''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#98ff98;" |'''R''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> ! <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#ADD8E6;" |'''N''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !73 |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#98ff98;" |'''R''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#ADD8E6;" |'''N''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#98ff98;" |'''R''' <!-- 23 --> |style="background:#98ff98;" |'''R''' <!-- 19 --> |style="background:#ADD8E6;" |'''N''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#98ff98;" |'''R''' <!-- 37 --> |style="background:#98ff98;" |'''R''' <!-- 41 --> |style="background:#ADD8E6;" |'''N''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> ! <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !89 |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#98ff98;" |'''R''' <!-- 71 --> |style="background:#98ff98;" |'''R''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#ADD8E6;" |'''N''' <!-- 43 --> |style="background:#ADD8E6;" |'''N''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#ADD8E6;" |'''N''' <!-- 3 --> |style="background:#98ff98;" |'''R''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#98ff98;" |'''R''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#ADD8E6;" |'''N''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> ! <!-- 89 --> |style="background:#98ff98;" |'''R''' <!-- 97 --> |- !97 |style="background:#ADD8E6;" |'''N''' <!-- 83 --> |style="background:#98ff98;" |'''R''' <!-- 79 --> |style="background:#ADD8E6;" |'''N''' <!-- 71 --> |style="background:#ADD8E6;" |'''N''' <!-- 67 --> |style="background:#ADD8E6;" |'''N''' <!-- 59 --> |style="background:#98ff98;" |'''R''' <!-- 47 --> |style="background:#98ff98;" |'''R''' <!-- 43 --> |style="background:#98ff98;" |'''R''' <!-- 31 --> |style="background:#ADD8E6;" |'''N''' <!-- 23 --> |style="background:#ADD8E6;" |'''N''' <!-- 19 --> |style="background:#98ff98;" |'''R''' <!-- 11 --> |style="background:#ADD8E6;" |'''N''' <!-- 7 --> |style="background:#98ff98;" |'''R''' <!-- 3 --> |style="background:#ADD8E6;" |'''N''' <!-- 5 --> |style="background:#ADD8E6;" |'''N''' <!-- 13 --> |style="background:#ADD8E6;" |'''N''' <!-- 17 --> |style="background:#ADD8E6;" |'''N''' <!-- 29 --> |style="background:#ADD8E6;" |'''N''' <!-- 37 --> |style="background:#ADD8E6;" |'''N''' <!-- 41 --> |style="background:#98ff98;" |'''R''' <!-- 53 --> |style="background:#98ff98;" |'''R''' <!-- 61 --> |style="background:#98ff98;" |'''R''' <!-- 73 --> |style="background:#98ff98;" |'''R''' <!-- 89 --> ! <!-- 97 --> |- |}
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