Friedman number

In base ${\displaystyle b=mk-m}$,

${\displaystyle b^{2}+mb+k=(mk-m+m)b+k=mbk+k=k(mb+1)}$

is a Friedman number (written in base ${\displaystyle b}$ as 1mk = k × m1).[4]

In base ${\displaystyle b>2}$,

${\displaystyle {(b^{n}+1)}^{2}=b^{2n}+2{b^{n}}+1}$

is a Friedman number (written in base ${\displaystyle b}$ as 100...00200...001 = 100..001², with ${\displaystyle n-1}$ zeroes between each nonzero number).[4]

In base ${\displaystyle b={\frac {k(k-1)}{2}}}$,

${\displaystyle 2b+k=2\left({\frac {k(k-1)}{2}}\right)+k=k^{2}-k+k=k^{2}}$

is a Friedman number (written in base ${\displaystyle b}$ as 2k = k²). From the observation that all numbers of the form 2k × b²ⁿ can be written as k000...000² with n 0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long. For example, for ${\displaystyle k=5}$, or in base 10, 250068 = 500² + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099 in base 10.[4]

Repdigit Friedman numbers:

• The smallest repdigit in base 8 that is a Friedman number is 33 = 3³.
• The smallest repdigit in base 10 that is thought to be a Friedman number is 99999999 = (9 + 9/9)^9−9/9 − 9/9.[4]
• It has been proven that repdigits with at least 22 digits are nice Friedman numbers.[4]

There are an infinite number of prime Friedman numbers in all bases, because for base ${\displaystyle 2\leq b\leq 6}$ the numbers

${\displaystyle n\times 10^{1111}+11111111=n\times 10^{1111}+10^{1000}-1+0+0}$ in base 2

${\displaystyle n\times 10^{102}+1101221=n\times 10^{102}+2^{101}+0+0}$ in base 3

${\displaystyle n\times 10^{20}+310233=n\times 10^{20}+33^{3}+0}$ in base 4

${\displaystyle n\times 10^{13}+2443111=n\times 10^{4+4}+(2\times 3)^{11}}$ in base 5

${\displaystyle n\times 10^{13}+25352411=n\times 10^{2\times 5-1}+(5+2)^{(3+4)}}$ in base 6

for base ${\displaystyle 7\leq b\leq 10}$ the numbers

${\displaystyle n\times 10^{60}+164351=n\times 10^{60}+(10+4-3)^{5}+0+0+\ldots }$ in base 7,

${\displaystyle n\times 10^{60}+163251=n\times 10^{60}+(10+3-2)^{5}+0+0+\ldots }$ in base 8,

${\displaystyle n\times 10^{60}+162151=n\times 10^{60}+(10+2-1)^{5}+0+0+\ldots }$ in base 9,

${\displaystyle n\times 10^{60}+161051=n\times 10^{60}+(10+1-0)^{5}+0+0+\ldots }$ in base 10,

and for base ${\displaystyle b>10}$

${\displaystyle n\times 10^{50}+{\text{15AA51}}=n\times 10^{50}+(10+{\text{A}}/{\text{A}})^{5}+0+0+\ldots }$

are Friedman numbers for all ${\displaystyle n}$. The numbers of this form are an arithmetic sequence ${\displaystyle pn+q}$, where ${\displaystyle p}$ and ${\displaystyle q}$ are relatively prime regardless of base as ${\displaystyle b}$ and ${\displaystyle b+1}$ are always relatively prime, and therefore, by Dirichlet's theorem on arithmetic progressions, the sequence contains an infinite number of primes.