Two new search strategies to find special-form primes p

[tevador]

This PR adds two new search strategies that look for special-form primes p with faster modular arithmetic. This is important for projects that will use the curve Ep much more often than curve Eq, such as Monero.

The new stategies work by enumerating special-form primes p and then solving for T and V using the Cornacchia's algorithm. If a solution exists and any of the 6 possible orders are prime, a curve cycle is guaranteed to exist.

Because special-form primes are rare, there is a new option --anyeqn that will find the smallest b such that both curves have the equation y^2 = x^3 + b.

Additional small changes are:

1. Primes are printed in hex format, which is arguably more readable.
2. Pool termination is handled correctly, so the script will exit if the search space is exhausted.

Crandall primes

Crandall primes have the form of 2^x-c with c small. The most famous example is the prime 2^255-19 used by Curve25519. These primes allow a fast Barrett modular reduction algorithm.

Unfortunately, this strategy does not work very well with the 2-adicity requirement, but can still be used to find cycles without it. For example:

sage amicable.sage --sequential --ignoretwist --anyeqn --crandall 255 1

The first cycle found is:

p   = 0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffad8b (bitlength 255, weight 248, 2-adicity 1)
q   = 0x800000000000000000000000000000014c6ce9c4bd00be001b8b4bcb29ef7267 (bitlength 256, weight 65, 2-adicity 1)
Ep/Fp : y^2 = x^3 + 23
Eq/Fq : y^2 = x^3 + 23

Montgomery-friendly primes

Primes in the form of c*2^x+1 with c small have a much more efficient Montgomery reduction algorithm. A big advantage is that these primes are naturally 2-adic.

Cycles found with this strategy always have 2-adicity of the prime p twice higher than the 2-adicity of prime q.

For example, the command:

sage amicable.sage --sequential --ignoretwist --anyeqn --montgomery 255 112

wil perform an exhaustive search and find a total of 5 cycles. The first cycle found is:

p   = 0x5eca430000000000000000000000000000000000000000000000000000000001 (bitlength 255, weight 13, 2-adicity 232)
q   = 0x5eca43000000000000000000000000010dd00000000000000000000000000001 (bitlength 255, weight 20, 2-adicity 116)
Ep/Fp : y^2 = x^3 + 278
Eq/Fq : y^2 = x^3 + 278

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All the changes are backwards compatible. The Pasta cycle can still be reproduced using the same command:

sage amicable.sage --sequential --requireisos --sortpq --ignoretwist --nearpowerof2 255 32