Large solutions to Pell's Equation

[Haydon Berrow]

@James

You have observed that Pell's Equation appears to have large solutions when N-1 has only small factors and most of the time I think you are right. If you were to plot a point-graph to illustrate this with the vertical axis as the (log of) the the solution, what would you choose as the horizontal axis?

Possibilities I have thought of so far are

• the cototient of N-1
• the largest prime in the factorisation of N-1
• the sum of the powers in the factorisation of N-1

I'm a bit concerned that more numbers will have small factors than have large factors and it can can only be expected that the larger set has the larger solutions.

Here are some big solutions that I found, the first three are prime, the fourth isn't

• 2³.5.19+1 = 761 has solution (1280001, 46400)
• 2³.3.5.19+1 = 2281 has solution (2981649830743654269669907724743299862819650512057127554474550049, 62430094928777423119382521794712492545154361356761603591711480)
• 2².3.149+1 = 1789 has solution (13673687937600285436522338047798889300505982960692087644059539022368201, 323281233024712565221770716009156212819348235266978239649677477568260)
• 2.3.139+1 = 835 has solution (34336355806, 1188258591)

[Haydon Berrow]

Addendum.

And this morning I'm not so sure.

Would you like to predict whether a Safe Prime (p is a Safe prime if (p-1)/2 is also prime) will be associated with a large or small solution to Pell's Equation? Ditto for prime numbers such that (p-1)/2 is a product of 2 primes?

[Haydon Berrow]

I have written a short article on this subject at <http://tumuluspi.blogspot.co.uk/2018/01/some-observations-on-large-solutions-to.html>