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.

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* According to an idea promoted by my friend Jaime Ayala, by large a devotee to Fermat.

Issue 1: Find the least factor of the Fermat number F₄₀₀.

The closest partially factorized Fermat number to F₄₀₀  is F₃₉₈ whose least factor (120845*2^401+1) was found by Wilfrid Keller in 1984.

Maybe you can find the correspondent least factor of  F₄₀₀ 

Suggestion: use the very efficient Leonid Durman's code

Issue 2: Find solutions to 2^x = 3 mod x

The least number x satisfying the given congruence is 4700063497,found by Lehmer & Lehmer (Ribenboim, p. 118). One more solution (but presumably not the next one) was found by Joe K. Crump: 8365386194032363, the 18/9/2000.
   

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)

Time ago I observed that 351725765537 and 655372571753 are both primes and reported this curio to my friend G.L.Honaker.

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 [N-1, Brillhart - Lehmer - Selfridge], arithmetic expression used: 10^((k-23)/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:

I believe that this falls under simple "chance". Keep in mind that simply
looking at values in the upper 30 mod (100) values (71 to 100), there is 1
value in the 70's not yet present (74) 2 values in the 80's not yet present
(83 and 86) and 3 values in the 90's (95 97 and 100) I certainly do not
see anything "overly" suspicious about m == 0 mod(100) other than F100, F200, ..., F1000 may have no small factors. When the Fermat number gets large, all bets are off at finding a "certain" Fermat factor. Any value over about a few hundred, and the search actually takes on a "strip mine" look, to where you simply start testing a large enough range of k's and n's until a factor drops out somewhere.

This is all simply a WAG from me (Wild &$$ Guess). It certainly could
be that all m==0 mod(100) are prime, but I highly doubt that ;) I certainly
am not the expert here, I simply have a dogged determination (along with a
little luck :)

On October 2005 finally we receive the following happy new:

My name is Asko Vuori, I live in Finland.
Today 29.sep.2005 my laptop (Celeron M 1400) with WinXP sp2 and
Fermat.exe v4.4 has found new Fermat divisor
 
found m=600, k=6213186413, n=605
6213186413.2^605+1 divides F_600

 

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
b) and argument by John Cosgrave about the possibility that exist another Fermat prime number

On January 2007, Joe Crump updated the search for the Issue 2: