Problems & Puzzles: Puzzles Puzzle 149. Fermat 400 This Friday 17, August 2001, Fermat's birthday reached to be 400 years old. Honoring this milestone date the current puzzle will deal* with three issues around some of the Fermat's numeric ideas, expecting not only solutions to these issues but that the reader contribute sending other issues in the same trend to remember this important French mathematician. ____ Issue 1: Find the least factor of the Fermat number F_{400}.
Maybe you can find the
correspondent least factor of F_{400 } Suggestion: use the very efficient Leonid Durman's code Issue 2: Find solutions to 2^{x} = 3 mod x
Find three more solutions to 2^{x} = 3 mod x, 4700063497<x<8365386194032363, or prove that there are not any more solutions in the given range Issue 3: Find nice primes concatenating aesthetically the five known Fermat primes (3, 5, 17, 257 & 65537)
Maybe this is the proper time to find some nice titanic primes concatenating someway aesthetically the five known Fermat primes Note: I'm expecting prime numbers composed ONLY by the five Fermat primes without one exception; something like the following one: 73556752715(3)97951725765537, C. Rivera, 18/8/2001, SPSP, Palindrome 1001 digits. Total digits a palindrome. Digits of the central nut also a palindrome. Found w/Ubasic. Solution: 1. r(89,175325765537)*1000+553 = (175325765537)89335 , Jim Fougeron, 18/8/2001, 1071 digits, Prime certified w/Titanix V2.1, 2h 28m on a PC Athlon 750. 2. 1(0)z35172576553735567527153(0)z1, C. Rivera, 19/8/2001, titanic palprime for k digits = 2.z+25 = 1321, 2341, 2691, 4575, 7523, next?, pfgw, certified by [N1, Brillhart  Lehmer  Selfridge], arithmetic expression used: 10^((k23)/2)*(10^((k+21)/2)+35172576553735567527153)+1
And your suggested issue is? Issue 4. Alberto Hernández  from Monterrey, México  points out that no Fermat number Fm has been factorized if m = 0 mod 100. Is this casual or interesting? Jim Fougeron got two prime Fermat factors this month in a lapse on no more than 10 days. So it was very natural for me to ask him what does he think about the Hernández's observation. This is his answer:
On October 2005 finally we receive the following happy new:
Leonid Durman has been adding new material related to the conjectured compositeness of the Fermat numbers Fm for m>4. For organization reasons these contributions have been added to the corresponding page in this site: Conjecture 4. We remind you to take a look at these interesting contributions. There you will se: a) an argument by Durman to bound the value of
the least prime factor of a Fermat number Fm On January 2007, Joe Crump updated the search for the Issue 2:






