# Tagged Questions

Questions on the number-theoretic functionality of Mathematica.

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### Factor a polynomial Root into Roots of smallest possible degree

Suppose I have a polynomial Root representing an algebraic number. I want to represent it (if possible) as a product of several polynomial ...
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### Finite Field matrix rank calculation

How does one define a matrix over $\mathrm{GF}(p^r)$ in Mathematica in order to compute rank? I am working with $\mathrm{GF}(2)$?
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### Simplifying expressions involving Divisible

FullSimplify[ Divisible[p^2 - 1, 24] , Element[p, Primes] && p > 3] Should evaluate to True, but I get ...
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### Are there any Mathematica programs on or about the LMFDB Archive?

I would like to explore the LMFDB Archive with L-functions using Mathematica. Is there any Mathematica sample code available to get me started?
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### FrobeniusSolve: how does it work?

Can someone suggest any reference to read? I would like to understand how the algorithm works.
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### Parallel PowerMod

Is there anyway to parallelize the PowerMod function? Here is my Left-To-Right modular exponentation: ...
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### Rationalize error

The docs state that "Rationalize[x,dx] yields the rational number with smallest denominator that lies within dx of x." However, testing this out it appears to be false. ...
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### Check Zagier theorem about Mahler's measure

I want to check the following theorem by using Mathematica: (from Heights of Polynomials and Entropy in Algebraic Dynamics, page 22) $\textbf{Theorem}.$ Let $\omega$ denote a primitive $6th$ ...
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### Undocumented function SumOfSquaresReps

There is an interesting (and documented) number-theoretic function in MMA called PowersRepresentations[$n$, $k$, $p$]. It gives the distinct representations of the integer $n$ as a sum of $k$ non-...
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### PowersRepresentations Algorithm

I'm trying to understand the mathematics behind counting the number of representations of a positive integer by $n$ distinct $k$th powers, i.e. I would really like to know how to do the Mathematica ...
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### Working with large integers in Modular Arithemetic

What would be the most efficient method of finding the remainder of the following division, 2^(2^330000000 - 1)/(2^330000001 - 1). I have tried using PowerMod function in Mathematica but i did not get ...
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### Coppersmith's algorithm like Pari's zncoppersmith?

Is there some Mathematica package (or built-in that I missed) available, more or less equivalent to Pari's zncoppersmith function? Paraphrasing that source: given ...
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### Elliptic curve cryptography in Mathematica

I can find no resources for doing elliptic curve cryptography. I have used the finite field package, but I find it cumbersome and it does not seem to have any builtin methods for ECC. How can I get ...
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### Montgomery Modular reduction

I saw a Mathematica Implementation of Montgomery reduction at Montgomery Modular Exponentiation, a beautiful one by Simon Woods. I ran this in Mathematica and it was fast, but it does not give the ...
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### Efficiently create a list of factors of consecutive integers

I'm interested in a scalable (read: sublinear) algorithm for producing the list of integer factors of each integer from 1 to n. ...
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### Factor Integer function with variable arguments

I'm trying to build a function that gives the highest power of a prime factor of a number. The following works perfectly: ...