# Tag Info

## Hot answers tagged computational-number-theory

Accepted

### Does any cubic polynomial become reducible through composition with some quadratic?

You should refer to Lemma 10 (page-233) in this paper by Schinzel where he proves that for any polynomial $F(x)$ of degree $d$ we have a polynomial $G(x)$ of degree $d-1$ such that their ...
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### Non-trivial solutions for $-(c^2-d^2)(a^2-b^2)=2(ad-bc)(bd+ac)$?

There is no such solution. Let $$Q(a,b,c,d) = 2(ad-bc)(bd+ac) + (a^2-b^2)(c^2-d^2)$$ be the difference between the two sides of the equation, so we seek to solve $Q(a,b,c,d) = 0$. This is a ...
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### What computer program for automorphic forms

The only CAS's that have built-in support for modular and automorphic forms, as far as I know, are Sage and Magma. [Edit: I had forgotten Pari/GP, which will introduce substantial modular forms ...
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### Possible contemporary improvement to bounded gaps between primes?

I think that there is indeed some possibility to lower the bound, and this is something I've looked at seriously a few times. I spent a semester (in 2019) with the Computational Number Theory Group ...
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### Question on the 50th (known) Mersenne prime number

As Jan Grabowski notes, all the primes that you mention have been discovered by GIMPS. GIMPS draws a distinction between testing and double-checking. When a Lucas–Lehmer test is performed on a ...
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### simple conjecture on palindromes in base 10

Using $$\frac{10^c-1}{9}=\sum_{m=0}^{c-1} 10^m,$$ the product in question equals $$\sum_{n=0}^{2a+2b+c-1}r(n)10^n,$$ where $r(n)$ is the number of times $n$ occurs among the numbers (counted with ...
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### Representing field elements in a computer

You are looking at a computable field (if your focus is on the field), or a computable presentation of a field (if your focus is on the details of how elements and operations are coded). These objects ...
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### Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$

Answer to Question 1. Yes, $E(\mathbb{Z})=F$. The inclusion $F\subseteq E(\mathbb{Z})$ follows from $E\subseteq E(\mathbb{Z})$ and Dickson's result $E=F$. It remains to show $E(\mathbb{Z})\subseteq F$...
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### Computing an eigencuspform in $S_2(\Gamma_0(1776))$

I did the computation in Sage, and there is no such form $f$. There are 21 Galois orbits of newforms of level $\Gamma_0(1776)$ and trivial character, of which the largest has size 3, and none of the ...
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### Goldbach conjecture and other problems in additive combinatorics

It seems what you are asking is "If we have a precise asymptotic for the number of elements of a set, can we solve binary additive problems involving that set?" The answer in general seems ...
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### Expressing primes $p\equiv 1 \pmod 3$ in the form $p = x^2 + xy + y^2$

This is an elaboration of the answer that Noam Elkies provided in the comments. Suppose that $p=x^2 + xy + y^2$. Then note that $x$ and $y$ are small relative to $p$ (at most half as many digits). ...

### Computation of a minimal polynomial

In Sage you can do something like: ...
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### Computing millions of coefficients of non self-dual modular forms

One shortcut you could use for computing the level 17 form you link to would be the following. There are exactly 8 Eisenstein series of weight 1 for $\Gamma_1(17)$ and they are all given by completely ...
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### Is there an efficient algorithm for finding a square root modulo a prime power?

An explicit formula is given in Tonelli's 1891 note referred to in the Wikipedia entry on the Tonelli-Shanks algorithm: Given a prime $p>2$ and a quadratic residue $a \bmod p$, let $x$ be a square ...

### existence of an elliptic curves with given number of points over finite field

Deuring proved that for every $a, |a| < 2\sqrt{p}$, there exists an elliptic curve with $p+1-a$ points over $\mathbb{F}_p$. M Deuring, Die Typen der Multiplikatorenringe elliptischer ...
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Yes. What you ask about are Stirling numbers of the first kind $s(j,j-k)$. Formula (21) http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html is an explicit expression for fixed $k$.
I want to leave a few elementary comments, maybe they will be helpful. The question asks about recurrence relation $$u_{n+1}= \lfloor u_n \rfloor (u_n − \lfloor u_n \rfloor + 1)$$ Suppose you write ...