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Why is integer factoring hard while determining whether an integer is prime easy?

What I think you're asking for are examples of search problems that seem to be hard, while a corresponding decision problem is solvable in polynomial time (but not totally trivial). It is true that ...
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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 ...
• 1,032
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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 ...
• 76.6k

Why is integer factoring hard while determining whether an integer is prime easy?

In the particular case of primality testing, primality testing is easier than factoring mainly because $p$ is prime if and only if $\mathbb{Z}/p\mathbb{Z}$ forms a field. This has a lot of ...
• 5,334

Primality of a number of more than 50k digits

Yes, it is possible, but it is close to the boundary of what is reasonable. See for instance this software that was recently used by Andreas Enge to prove the primality of $10^{50000}+65859$. It took ...
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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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Why is integer factoring hard while determining whether an integer is prime easy?

Let me give a slightly different example. The Baumslag-Gersten group is a one-relator group with the presentation $\langle a, b \mid [a, a^b] = a\rangle$. This has a Dehn function which grows faster ...
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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 ...
• 17.6k
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• 64.8k
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Can we compute the first $n$ digits of $\pi$ in $F(n)$ time?

Mahler proved(1) that for any $p,q$, $\left | \pi -\frac{p}{q} \right| > \frac{1}{q^{42}}$. It follows that if one can compute $2^{ 42n} \pi$ to within an error of at most $1$, one can compute the ...
• 131k
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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 ...
• 76.8k
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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 ...
• 94.9k
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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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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 ...
• 11.7k
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Parity of number of solutions to Diophantine equations

While it’s unclear what you mean by “parity” when the number of solutions is infinite, all such questions have been answered by M. Davis, On the number of solutions of Diophantine equations, Proc. AMS ...
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A little number theoretic game

(This is not an answer, but an extensive comment and numerical simulation about Grundy values.) I believe there is some level of confusion because there are actually two very similar games under ...
• 28.4k
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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). ...

A little number theoretic game

Edit: I added code to compute the Nim values for the first $N$ positions of this game after the original post, as requested by @Timothy-Chow. Unfortunately my results don't match those given by @Peter-...
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Can every integer be written as a sum of squares of primes?

The answer is yes, and this follows from known results concerning the Waring-Goldbach problem.
• 94.9k

Computation of a minimal polynomial

In Sage you can do something like: ...
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In which cyclic cubic number fields does there exist this type of unit?

The answer to question 1 is yes. Pick any field element $x_1$ outside the rationals. Let $x_2$ and $x_3$ be its conjugates under the Galois group. Then the cross ratio w=\frac{(x_1-1)(x_2-x_3)}{(...
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Fermat's last theorem $\pm1$

[EDITED] It is likely that there are no solutions at all for $n \ge 4$. For $n \ge 5$ a solution would be a counterexample to the Lander, Parkin, and Selfridge conjecture. The best FLT "near ...
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An efficient isomorphism between finite fields

Google provides an answer to this question. The first deterministic polynomial time algorithm for this is due to H. W. Lenstra, Jr., in his paper "Finding isomorphisms between finite fields" (...
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• 26.3k