```Date: Dec 6, 2012 6:38 PM
Author: Dr J R Stockton
Subject: Re: No Putnam spoilers please

In sci.math message <Pine.NEB.4.64.1212041922190.1746@panix1.panix.com>,Tue, 4 Dec 2012 19:48:01, William Elliot <marsh@panix.com> posted:>On Tue, 4 Dec 2012, Dr J R Stockton wrote:>> Sun, 2 Dec 2012 19:47:28, William Elliot <marsh@panix.com> posted:>> >On Sat, 1 Dec 2012, William Elliot wrote:>>>> >> "What's 5^2012 mod 7?" is a Putnam question? ;-}>>> >phi(7) = 6;  5^2012 = 5^(6 * 335 + 2) = 5^2 = 25 = 4 (mod 7)>>>> Correct by direct calculation in Bases 7, 10, 13, using my longcalc.exe.>>For all n in N, 5^n = 5 (mod 10)>Proof by induction.  5^1 = 5 (mod 10).>If 5^n = 5 (mod 10), then 5^(n+1) = 5^n * 5 = 5 * 5 = 25 = 5 (mod 10)>>phi(13) = 12;>5^2012 = 5^(12 * 168 + 6) = 5^6 = 25^3 = (-1)^3 = -1 = 12 (mod 13)>>1200> 720>  96>>longcalc.exe is too smart for it's britches.In that calculation, longcalc uses only elementary arithmetic, as usedto be taught in schools in my day.  Your proof, however, is 50%incomprehensible to me.  Though it may well be right.  Longcalc found anerror in *a* printed representation of (3^349-1)/2, which you might havedifficulty with. Fx : checks : I think it has found another one.--  (c) John Stockton, Surrey, UK.  ?@merlyn.demon.co.uk   Turnpike v6.05   MIME.   Web  <http://www.merlyn.demon.co.uk/> - FAQish topics, acronyms, & links. Proper <= 4-line sig. separator as above, a line exactly "-- " (SonOfRFC1036) Do not Mail News to me. Before a reply, quote with ">" or "> " (SonOfRFC1036)
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