Template:Sequence of the Day for July 6

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Intended for: July 6, 2012

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First draft entered by Alonso del Arte on March 7, 2011 based on an almost verbatim copy of a write-up by David W. Wilson from October 23, 2010. ✓
Draft reviewed by Alonso del Arte on March 24, 2012 ✓
Draft approved by June 6, 2012

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A036236: Least inverse of A015910: smallest integer $\scriptstyle k \,>\, 0 \,$ such that $\scriptstyle 2^k \bmod k \,=\, n,\, n \,\ge\, 0, \,$ or 0 if no such $\scriptstyle k \,$ exists.

{ 1, 0, 3, 4700063497, 6, 19147, 10669, 25, ... }

Fermat's little theorem says that $\scriptstyle {2^p} \,\equiv\, 2 \pmod{p} \,$ for odd prime $\scriptstyle p \,$ , and on some random day long ago I got to wondering what values $\scriptstyle {2^n} \bmod n \,$ might take for other $\scriptstyle n \,$ . I did some experimenting and found that that for small $\scriptstyle n \,$ , $\scriptstyle {2^n} \bmod n \,$ took on many small values, but 1 and 3 remained elusive. I asked about them on the seqfan list, and found that $\scriptstyle {2^n} \bmod n \,=\, 1 \,$ is provably insoluble, while D. H. Lehmer had found the impressively large minimal solution $\scriptstyle n \,$ = 4700063497 for $\scriptstyle {2^n} \bmod n \,=\, 3 \,$ , an impressively large solution for such a simple identity.

Joe Crump has done some amazing work with this sequence, see 2^n mod n = c.

Template:Sequence of the Day for July 6

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