RFI… Can Any Number Theorist (or others) Reply to This?

I can’t answer the following for one of my readers, but maybe someone else can (about a Fermat factoring variant)???

Well over 3 years ago I posted this little gem from Futility Closet:

"In 1643, Marin Mersenne wrote to Pierre de Fermat asking whether 100895598169 were a prime number.

Fermat replied immediately that it's the product of 898423 and 112303, both of which are prime.

To this day, no one knows how he knew this. Has a powerful factoring technique been lost?"

At the time, one comment came in (from someone named Walt), as follows:

“It is not that surprising if we assume that Fermat used what we call Fermat's factoring method. Upon multiplying by the cofactor of 8 (and using difference of pronic numbers instead of squares since the number is now even) he would find this pair of factors on the first try. (I am assuming that taking the square root of a 12-digit number was feasible. Also, I have no idea how difficult it was at that time for him to prove the primality of the resulting 6-digit factors.)
In fact, 8 * 112303 = 898424 = 1 + 898423. This is a remarkable coincidence and makes me wonder if Mersenne used this relation to construct the problem in the first place.”

Now (3+ years later) I’ve received an inquiry from a retired German mathematician, who recently ran across the post & comment, and asking in part about the:

“…. variant of ‘what we call Fermat's factoring method. Upon multiplying by the cofactor of 8 (and using difference of pronic numbers instead of squares since the number is now even) …’  I am wondering about (t)his remark to use differences of pronic numbers: I have never heard of that variant or read about it in any book. He [Walt] does not quote any references, so he seems to consider this variant as being well known. In fact, it is not too difficult to figure out the formula a*(a+1) - b*(b+1) = (a+b+1)*(a-b) that turns such a difference into a product, and to check that one may thus devise a factorization method for even numbers. I am wondering if this has been published anywhere.”

Can anyone provide an answer???

Get A Life!

In his latest book, "Numbers: Their Tales, Types, and Treasures," Alfred Posamentier mentions what he labels, "Pythagorean Curiosity #4":

It seems that in the mid-1600s the ever-inquisitive Pierre de Fermat sought a Pythagorean triple wherein the SUM of the two smaller values (a + b) was a square integer, AND the largest triple (c) was also a square integer.
Well, he found one such triple:

(a) 4,565,486,027,761
(b) 1,061,652,293,520 and
(c) 4,687,298,610,289

where a + b = 5,627,138,321,281 or 2,372,159²  and c = 2,165,017² 

Mind you, of course, no computers in those days!

MOREOVER, Fermat proved that this was the smallest such Pythagorean triple! (I don't know if any more such triples have been found in the almost four centuries since?)

All of which leads me to imagine being alive in 1643 (when Fermat concocted the problem) and sayin', "YO Pierre, uhhh, GET A LIFE!"  ;-)

Fermat's Last Theorem (VIDEO)

"Math-Fail" directs readers to view this video on Andrew Wiles and Fermat's Last Theorem. It is indeed excellent and enthralling (45 mins. long). I too encourage all math-lovers who have never seen it to find the time to view it. It very well depicts the love, devotion, and elation a human being can experience toward a subject that so many others view as merely dry and tedious:



(...and what I really like is Wiles' workdesk, which actually makes mine seem almost orderly! ;-))

A Look At Pascal's Triangle

In his book "Wonder of Numbers" Clifford Pickover names the following article as having the "all-time strangest title" of any published mathematical paper:

Granville, A. (1992) "Zaphod Beeblebox's brain and the fifty-ninth row of Pascal's Triangle" American Mathematical Monthly April, 99(4): 318-331.

The paper (pdf) can be found here:

http://www.gianpierobiancoli.it/wp-content/uploads/2009/10/beeb.pdf

And if you don't know who the character Zaphod Beeblebrox is (from "Hitchhiker's Guide to the Galaxy") you can check him out here:

http://en.wikipedia.org/wiki/Zaphod_Beeblebrox

In other matters, a quickie intro to the Riemann Hypothesis from Matt Parker here:


http://tinyurl.com/2arhu4r 

[includes the mention that "All prime numbers (greater than five) squared are one more than a multiple of 24."]

And revisiting Mr. Fermat:

x^n + y^n = z^n  ...NO solutions for n ≥ 3

x^n + y^n = z^(n-1) ...INFINITELY many solutions for all n ≥ 3

proof:

http://tinyurl.com/2d5qucy