# [ANN] euler, an arithmetic library for native integers

In case anyone is interested, I let you know of `euler`, a small arithmetic library of mine. Unfortunately it is not published on opam (but there is an opam file), and I won’t be working on it anymore, so anyone is welcome to take over the project or just steal code from it. If you see fun in it, I left plenty of TODOs!

The library is documented, with a focus on algorithmic complexities, and implementation code has a lot of comments too.

repo, doc preview.

# What’s in it?

`euler` is a library for doing integer arithmetic with OCaml’s standard integers (31 or 63 bits). It provides:

• Drop-in, overflow-detecting base arithmetic:
if you are paranoid about vicious bugs sneaking in silently, this library detects overflows and signal them by throwing an exception; the module can be used as a drop-in replacement for the standard library (beware that `Euler.Arith.min_int` differs from `Stdlib.min_int`, the latter being a forbidden value). There are also a few additional functions such as integer logarithms and square roots.
for the weird folks (like myself) who are interested in advanced arithmetic but do not care about integers larger than 262, and thus do not want the burden of using an arbitrary-precision library (zarith of GMP), there you are. The library provides some classic functions such as
• the GCD,
• the Jacobi symbol,
• primality testing (fast and deterministic for all 63-bit integers!),
• integer factorization (implementing Lenstra’s elliptic curve factorization, which was apparently one of the best known algorithms back when I wrote that code, but obviously it is still very slow! — and I must say I understand very little about it…),
• a prime sieve (heavily optimized) and a factorization sieve,
• Euler’s totient function (slow too, of course),
• and so on.
• Solvers for some forms of integer equations (so-called “Diophantine equations”):
• linear congruence systems (the Chinese remainder theorem),
• Pell-Fermat’s equations (the Chakravala method) — preliminary code that just needs some packaging effort).
• Modular arithmetic:
including finding modular inverses (and pseudo-inverses). A nice functorial interface provides convenient notations and uses a private type to enforce that values are always normalized in the range 0…m−1 where m is the modulus. Example use:
``````module M = Euler.Modular.Make (struct let modulo = 42 end)
let () = assert (M.( !:1 /: (!:33 +: !:4) = !:5 **:(-4) ))
(* said otherwise, modulo 42, the inverse of (33 + 4) is equal to 5^(−4) *)
``````

# But why?

Writing this library was fun and educative for me, and allowed me to solidify my math training in code. In fact, as the name suggests, the initial incentive was playing with Project Euler (hence the focus on integers that fit in a machine word) while sparing me the boredom of implementing a prime sieve for the hundredth time.

Nevertheless, I believe `euler` might prove actually useful outside of the playground. Overflow detection is an actual need in some software, and implementing it is not trivial, even less so after some amount of micro-optimization (see code). Modular arithmetic is not trivial either (e.g. multiplication is not as simple as `(a * b) mod m` because this computation might overflow). And well, even integer logarithms and square roots are handy at times, and again they not trivial to implement (as using their floating-point counterpart gives incorrect results for large integers).

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It might still be worth making one release on opam-repository such that it could be easily used, even if you won’t be updating it and making more releases.

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Can you upload the code on github?
I want to look at some code in there.

I have a copy forked here if you want: GitHub - mseri/euler-lib: Fork of https://gitlab.crans.org/mevel/euler-lib

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Great! You might want to pull again, as I just pushed two small commits (more TODOs!). And since the number of commits is now a satisfying 100, I think that’s all for me.

That I can do! So now the repo is there (argh, 101).

(It seems I am unable to edit my first post so as to put the new link, well.)

Ah fantastic, thanks! I have synced them again.

M

I would be happy to make a release on the opam-repository if you are fine with it. If you are comfortable with it, you could also give me access to your repo on GitHub and I can do it from there directly.

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Thanks to @mseri, `euler` is now on opam! And in the meantime it has gained more functions, the most striking of which being `Primes.prime_seq : int -> int Seq.t` (thank @cuihtlauac for pointing me to the algorithm!) and `Arith.smallest_root : int -> int * int` (the latter in the next release, 3.0, not on opam yet).