Editing Friedman number (section)

=== General results ===
In base <math>b = mk - m</math>, 
: <math>b^2 + mb + k = (mk - m + m)b + k = mbk + k = k(mb + 1)</math> 
is a Friedman number (written in base <math>b</math> as 1''mk'' = ''k'' × ''m''1).<ref name="erich">{{Cite web|url=https://erich-friedman.github.io/mathmagic/0800.html|title = Math Magic}}</ref>

In base <math>b > 2</math>,
: <math>{(b^n + 1)}^2 = b^{2n} + 2{b^n} + 1</math>
is a Friedman number (written in base <math>b</math> as 100...00200...001 = 100..001<sup>2</sup>, with <math>n - 1</math> zeroes between each nonzero number).<ref name="erich" />

In base <math>b = \frac{k(k - 1)}{2}</math>,
: <math>2b + k = 2\left(\frac{k(k - 1)}{2}\right) + k = k^2 - k + k = k^2</math>
is a Friedman number (written in base <math>b</math> as 2''k'' = ''k''<sup>2</sup>). From the observation that all numbers of the form 2''k'' × b<sup>2''n''</sup> can be written as ''k''000...000<sup>2</sup> with ''n'' 0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long. For example, for <math>k = 5</math>, or in [[base 10]], 250068 = 500<sup>2</sup> + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099 in [[base 10]].<ref name="erich"/>

[[Repdigit]] Friedman numbers:
* The smallest repdigit in [[base 8]] that is a Friedman number is 33 = 3<sup>3</sup>. 
* The smallest repdigit in [[base 10]] that is thought to be a Friedman number is 99999999 = (9 + 9/9)<sup>9−9/9</sup> − 9/9.<ref name="erich"/> 
* It has been proven that [[repdigit]]s with at least 22 digits are nice Friedman numbers.<ref name="erich"/>

There are an infinite number of prime Friedman numbers in all bases, because for base <math>2 \leq b \leq 6</math> the numbers
: <math>n \times 10^{1111} + 11111111 = n \times 10^{1111} + 10^{1000} - 1 + 0 + 0</math> in base 2
: <math>n \times 10^{102} + 1101221 = n \times 10^{102} + 2^{101} + 0 + 0</math> in base 3
: <math>n \times 10^{20} + 310233 = n \times 10^{20} + 33^{3} + 0</math> in base 4
: <math>n \times 10^{13} + 2443111 = n \times 10^{4 + 4} + (2 \times 3)^{11}</math> in base 5
: <math>n \times 10^{13} + 25352411 = n \times 10^{2 \times 5 - 1} + (5 + 2)^{(3 + 4)}</math> in base 6
for base <math>7 \leq b \leq 10</math> the numbers
: <math>n \times 10^{60} + 164351 = n \times 10^{60} + (10 + 4 - 3)^5 + 0 + 0 + \ldots</math> in base 7,
: <math>n \times 10^{60} + 163251 = n \times 10^{60} + (10 + 3 - 2)^5 + 0 + 0 + \ldots</math> in base 8,
: <math>n \times 10^{60} + 162151 = n \times 10^{60} + (10 + 2 - 1)^5 + 0 + 0 + \ldots</math> in base 9,
: <math>n \times 10^{60} + 161051 = n \times 10^{60} + (10 + 1 - 0)^5 + 0 + 0 + \ldots</math> in base 10, 
and for base <math>b > 10</math>
: <math>n \times 10^{50} + \text{15AA51} = n \times 10^{50} + (10 + \text{A}/\text{A})^5 + 0 + 0 + \ldots</math>
are Friedman numbers for all <math>n</math>. The numbers of this form are an arithmetic sequence <math>pn + q</math>, where <math>p</math> and <math>q</math> are relatively prime regardless of base as <math>b</math> and <math>b + 1</math> are always relatively prime, and therefore, by [[Dirichlet's theorem on arithmetic progressions]], the sequence contains an infinite number of primes.