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 ...
- 76.8k
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 ...
- 1,032
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 ...
- 76.6k
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 ...
- 5,334
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 ...
- 4,819
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 ...
- 35.7k
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 ...
- 17.6k
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
$$ 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, ...
- 106k
19
votes
Sum of squares and divisibility
This is not a complete answer, but just a way to transform the problem into one that can be attacked by brute force in some known way.
Write $d_i^2=N/n_i$. Then your relation becomes $$\frac{1}{n_1}+ \...
- 64.8k
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 ...
- 131k
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 ...
- 76.8k
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 ...
- 94.9k
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 ...
- 4,076
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 $E(\mathbb{Z})\subseteq F$...
- 2,560
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 ...
- 11.7k
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 ...
- 43.4k
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 ...
- 28.4k
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). ...
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-...
- 1,803
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.
- 94.9k
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)}{(...
- 2,222
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 ...
- 53.2k
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" (...
- 6,327
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 <...
- 43.2k
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 ...
- 29.9k
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 ...
- 38.4k
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 ...
- 26.3k
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 ...
- 19k
Only top scored, non community-wiki answers of a minimum length are eligible