55
votes

Accepted

### 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 ...

33
votes

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 ...

33
votes

Accepted

### 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 ...

26
votes

### 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 ...

23
votes

### 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 ...

21
votes

### 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 ...

21
votes

### 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 ...

20
votes

Accepted

### 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 ...

20
votes

Accepted

### Is there a way to specify a special kind of reciprocals of natural numbers?

This is a textbook example of a question for which one should turn to the OEIS for assistance. The first few elements of this set are
$$ 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ...

19
votes

### Sum of squares and divisibility

This is not a complete answer, but just a way to transform the problem into one that can be attacked by brute force in some known way.
Write $d_i^2=N/n_i$. Then your relation becomes $$\frac{1}{n_1}+ \...

18
votes

Accepted

### 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 ...

17
votes

Accepted

### 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 ...

16
votes

Accepted

### 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 ...

16
votes

Accepted

### 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 ...

16
votes

### 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 $E(\mathbb{Z})\subseteq F$...

16
votes

Accepted

### 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 ...

16
votes

Accepted

### 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 ...

16
votes

### 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 ...

15
votes

Accepted

### 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). ...

Community wiki

15
votes

### 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-...

15
votes

Accepted

### 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.

14
votes

14
votes

### 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)}{(...

14
votes

### 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 ...

13
votes

Accepted

### 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" (...

13
votes

Accepted

### Computational complexity of finding the smallest number with n factors

The problem asks for the least number $N$ such that the number of divisors of $N$ is at least $n+2$. Since all numbers below $N$ must have fewer divisors, clearly $d(N) > d(m)$ for all $1\le m <...

13
votes

### Twin primes conjecture and extrapolation method

This is just to present a numerical curiosity. I suspect that it is coincidental, but I feel compelled to share it in case anyone wants to look further.
First some disclaimers: Consider a random walk ...

13
votes

### Do consecutive integers have a big prime factor?

Shorey and Tijdeman, On the greatest prime factors of polynomials at integer points, Compositio Mathematica, tome 33, no 2 (1976), p. 187-195
http://www.numdam.org/item?id=CM_1976__33_2_187_0 note ...

13
votes

Accepted

### Computational efficiency of character sums for counting finite field points on a curve

For numbers as small as $5$, it should make virtually no difference. Let me interpret this question as asking for a method to do it over any finite field. I will also restrict to the case of $\mathbb ...

13
votes

Accepted

### On a fast high precision numerical analysis C library

Since you speak about mathematical proofs, probably you don't want an arbitrary-precision library, but a verified computation library based on interval arithmetic.
Maybe Arb? Or boost-interval?
And ...

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