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Quadratic residue
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===Prime power modulus=== All odd squares are β‘ 1 (mod 8) and thus also β‘ 1 (mod 4). If ''a'' is an odd number and ''m'' = 8, 16, or some higher power of 2, then ''a'' is a residue modulo ''m'' if and only if ''a'' β‘ 1 (mod 8).<ref>Gauss, DA, art. 103</ref> <blockquote>For example, mod (32) the odd squares are :1<sup>2</sup> ≡ 15<sup>2</sup> ≡ 1 :3<sup>2</sup> ≡ 13<sup>2</sup> ≡ 9 :5<sup>2</sup> ≡ 11<sup>2</sup> ≡ 25 :7<sup>2</sup> ≡ 9<sup>2</sup> ≡ 49 ≡ 17 and the even ones are :0<sup>2</sup> ≡ 8<sup>2</sup> ≡ 16<sup>2</sup> ≡ 0 :2<sup>2</sup> ≡ 6<sup>2</sup>≡ 10<sup>2</sup> ≡ 14<sup>2</sup>≡ 4 :4<sup>2</sup> ≡ 12<sup>2</sup> ≡ 16. </blockquote> So a nonzero number is a residue mod 8, 16, etc., if and only if it is of the form 4<sup>''k''</sup>(8''n'' + 1). A number ''a'' relatively prime to an odd prime ''p'' is a residue modulo any power of ''p'' if and only if it is a residue modulo ''p''.<ref name="Gauss, DA, art. 101">Gauss, DA, art. 101</ref> If the modulus is ''p''<sup>''n''</sup>, :then ''p''<sup>''k''</sup>''a'' ::is a residue modulo ''p''<sup>''n''</sup> if ''k'' ≥ ''n'' ::is a nonresidue modulo ''p''<sup>''n''</sup> if ''k'' < ''n'' is odd ::is a residue modulo ''p''<sup>''n''</sup> if ''k'' < ''n'' is even and ''a'' is a residue ::is a nonresidue modulo ''p''<sup>''n''</sup> if ''k'' < ''n'' is even and ''a'' is a nonresidue.<ref>Gauss, DA, art. 102</ref> Notice that the rules are different for powers of two and powers of odd primes. Modulo an odd prime power ''n'' = ''p''<sup>''k''</sup>, the products of residues and nonresidues relatively prime to ''p'' obey the same rules as they do mod ''p''; ''p'' is a nonresidue, and in general all the residues and nonresidues obey the same rules, except that the products will be zero if the power of ''p'' in the product β₯ ''n''. Modulo 8, the product of the nonresidues 3 and 5 is the nonresidue 7, and likewise for permutations of 3, 5 and 7. In fact, the multiplicative group of the non-residues and 1 form the [[Klein four-group]].
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