 mersenneforum.org > Math Mersenne prime in a Cunningham chain
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R.D. Silverman

Nov 2003

22·5·373 Posts Quote:
 Originally Posted by Dr Sardonicus According to the references in the link, the Diophantine equation 2n - 7 = x2 is called Ramanujan's square equation. He posed the question of whether it has any solutions for n > 15 in 1913. So I guess you're asking, did he understand the difference between math and numerology?
Yes. Ramanujan. But I saw no math related to this equation in the discussion.
All I saw was a lot of blind computation and sequences of numbers. Indeed, the
first use of the word "equation" came in your post.   2020-05-19, 08:01   #13
kruoli

"Oliver"
Sep 2017
Porta Westfalica, DE

251 Posts Quote:
 Originally Posted by JeppeSN The only pending is 2^74207282 - 3.
For the record, it has no factors smaller than $$2^{32}$$ and no factor found with P-1 using B1=100,000 and B2=5,000,000. I'll be running some ECM curves on it since it should be more efficient than on mersenne numbers.   2020-05-19, 13:53   #14
Dr Sardonicus

Feb 2017
Nowhere

22×72×17 Posts Quote:
 Originally Posted by R.D. Silverman Yes. Ramanujan. But I saw no math related to this equation in the discussion. All I saw was a lot of blind computation and sequences of numbers. Indeed, the first use of the word "equation" came in your post.
Here's some basic math regarding the equation 2n - 7 = x2. Rewriting as

x2 - 2n = -7

we have two cases:

x2 - y2 = -7 if n is even, and

x2 - 2*y2 = -7 if n is odd.

In both cases we want solutions where y is a power of 2.

In the first case, there is only one solution in positive integers x and y, x = 3 and y = 4.

In the second case, there are two sequences of solutions. Again, we want y to be a power of 2. The y-sequences may be described as

xn + yn*t = lift((2*t +/- 1)*Mod(3 + 2*t,t2 - 2)n), n = 0, 1, 2, ...

The y-sequences are

y0 = 2, y1 = 8, yk+2 = 6*yk+1 - yk (next terms 46, 268, 1562,...)

and

y0 = 2, y1 = 4, yk+2 = 6*yk+1 - yk (next terms 22, 128, 746,...)

Because of the factors 2*t +/- 1 in the explicit formula, the resulting sequences don't have the nice divisibility properties of the coefficients from the powers of a unit, so ruling out powers of 2 is correspondingly more difficult.   2020-06-15, 15:26 #15 JeppeSN   "Jeppe" Jan 2016 Denmark 2·71 Posts This may seem silly, but I ran a test with PFGW (slow, one CPU thread) on the remaining number. It took several weeks. Here is the result (PFGW does a 3-PRP test by default): Code: PFGW Version 4.0.1.64BIT.20191203.Win_Dev [GWNUM 29.8] Resuming at bit 72193575 2^74207282-3 is composite: RES64: [B486B8C605802535] (60360.4698s+0.0144s) (The 60k seconds is only for the part between iteration ~72M to ~74M.) So now the residue is here, if someone some day should want to repeat this long calculation. /JeppeSN   2020-06-15, 17:13   #16
R.D. Silverman

Nov 2003

22×5×373 Posts Quote:
 Originally Posted by JeppeSN This may seem silly,
We agree   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post firejuggler Math 31 2014-01-08 18:28 kosta Factoring 24 2013-03-21 07:17 Raman Cunningham Tables 32 2012-07-10 22:27 cipher Math 1 2009-09-01 15:12 olivier_latinne Math 54 2008-03-12 10:04

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