# Common Number Theory Algorithms in Python

My number theory library provides the following common number theory functions and algorithms:

 gcd greatest common divisor lcm least common multiple xgcd extended greatest common divisor cong solve a.x = p (mod n) for x mod_inverse inverse of a (mod n) crt chinese remainder theorem isqrt integer square root iroot integer k’th root miller_rabin Miller Rabin probable prime test Sieve a class for the sieve of Eratosthenes Primes a class for working with primes is_prime test if a positive integer is prime factor prime factors and their multiplicity factors prime factors (with repeated primes) divisors divisors of an integer divisor_pairs divisor pairs of an integer tau number of divisors of an integer omega sum of the divisors of an integer phi the Euler totient function lambdaf Carmichael function, min m for a^m == 1 (mod n) , all a co-prime to n mu the Mobius function jacobi the Jacobi symbol for an odd integer coprime_pairs Generate co-prime pairs pythag_primitive generate primitive Pythagorean triangles in order up to maximum side pythag generate all Pythagorean triangles in order up to maximum side rseq generate recursive sequences hensel_lift_2 Hensel root lifting for powers of 2 hensel_lift Hensel root lifting for powers of odd primes sqrt_mod_p the modular square root for a prime modulus sqrt_mod_pn the modular square root for prime power modulus sqrt_mod_m the modular square root for a composite modulus PQa continued fraction expansion of a quadratic irrational Qde class for solving x^2 – d.y^2 = n frobenius_number the Frobenius number for an integer sequence frobenius_solve Frobenius solutions for an integer sequence

Please note that these routines are still under development so they may contain bugs

My polynomial library provides facilities for manipulating polynomials with rational coefficients (this is used in the Funny Dice Sunday Times Teaser).

1. This is cool code. I finally got around to trying the Frobenius function and it works great!… as long as at least two of the items are coprime. When they all have a common factor > 1, the funcntion returns an answer that doesn’t make sense to me. As far as I can tell this is not the result of anything I did but I wouldn’t rule it out.

Here are a few examples of my program using your function:

d:dev>rpn [ 6 9 20 ] frobenius
43

d:dev>rpn [ 6 9 24 ] frobenius
3

d:dev>rpn [ 6 8 13 ] frobenius
23

d:dev>rpn [ 6 8 12 ] frobenius
10

I haven’t had a chance to go through the function to see if I can figure out why it does this. It will probably take me some time to understand it well enough, but I thought I would drop you a note.

p.s. I’m looking forward to implementing operators for some of the other functions as well.
Now I am off to McDonald’s to order 43 Chicken McNuggets!

• Frobenius numbers are only defined for integer sequenes for which:

${gcd}(a_1, a_2, …, a_n) = 1$

You can add a check by changing the last line in the frobenius_number function with the following:

Alternatively add this before the last line:

I have been updating other parts of the library today so I have added this check as well.

2. I figured if input with a gcd > 1 was disallowed for some reason, I would make the check myself and throw an error.

Thanks again. I still want to dig through and understand how it works. I’m still a number theory newbie.

Rick

3. The sqrtmod code is incorrect — specifically, for each m divisible by 3 or more distinct primes, sqrt_mod_m(a, m) returns incorrect results for a handful of values of a.

• Thanks for reporting the bug, which I have now fixed.

4. Also, could you explain the output of PQa()?

• There is a nice description here.

5. 6. 