Mersenne prime in a Cunningham chain

[R.D. Silverman]

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.

[kruoli]

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.

[Dr Sardonicus]

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

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

The y-sequences are

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

and

y₀ = 2, y₁ = 4, y_k+2 = 6*y_k+1 - y_k (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.

[JeppeSN]

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):

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

[R.D. Silverman]

Originally Posted by JeppeSN

This may seem silly,

We agree