If a3+b3+c3=Na^3+b^3+c^3=N, then x3+y3+z3+t3=Nx^3+y^3+z^3+t^3 = N in infinitely many ways?

It is well-known that, a3+b3+c3=N for N=1,2 is solvable in the integers in infinitely many ways . However, it is an open question (but is conjectured) that if for general N it has a solution, then it must have infinitely many. I’d like to propose a variant, Conjecture: If a3+b3+c3=N, then x3+y3+z3+t3=N in infinitely many … Read more

Short Diophantine definition of the sum-of-divisors function (using less than 100 variables)?

Is there a short Diophantine definition of the sum-of-divisors function? Is there a polynomial p such that c=∑d|nd ↔ ∃x1,…x100 p(c,n,x1,…x100)=0 ? This comes from a MathStackExchange post, where I suggested that standard algorithms would produce a polynomial in thousands of variables. Can we do significantly better, maybe with a bit more number theory? A short definition in Diophantine … Read more

Tripathi’s formulas for Frobenius number in three variables

Recently (January 2017) a paper by A. Tripathi has been published in the Journal of Number Theory with “closed” formulas for the Frobenius number in 3 variables (Formulae for the Frobenius number in three variables). Question: Have Tripathi’s formulas less computational complexity than already known algorithms (Greenberg, Rödseth) in some case? Answer AttributionSource : Link … Read more

Number of nontrivial integral solutions to f(x)=f(y)f(x)=f(y)

Let f(x)∈Z[x] be a nonconstant polynomial, and let g(x,y)=f(x)−f(y)x−y∈Z[x,y]. Let N(B) denote the number of pairs of integers (x0,y0) such that 1≤x0,y0≤B and g(x0,y0)=0. Is it possible to obtain explicit asymptotics for N(B) as B→∞? Answer AttributionSource : Link , Question Author : 352506 , Answer Author : Community

Is there a permutation $\pi\in S_n$ with $\sum\limits_{07$?

Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$. QUESTION: Is it true that for each $n=8,9,\ldots$ we have $$\sum_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0\tag{$*$}$$ for some $\pi\in S_n$? Let $s(n)$ denote the number of permutations $\pi\in S_n$ with $\pi(1)<\pi(n)$ satisfying $(*)$. Via Mathematica, I find that $$s(1)=s(2)=\ldots=s(7)=0,\ s(8)=1,\ s(9)=s(10)=4,\ s(11)=55.$$ When $n=8$ we can only take $$(\pi(1),\ldots,\pi(8))=(4,5,2,7,3,1,6,8)$$ … Read more

Limit of the real part of a geometric sequence

I came across the following problem, which turned out to be surprisingly hard: Show that lim Intuitively, setting z=\frac{1+i\sqrt{7}}{2}=\sqrt{2}e^{i\theta}, we see that |\mathrm{Re}(z^n)|=2^{\frac{n}{2}}|\cos (n\theta)|, so that as long as n\theta \ (\mathrm{mod}\ \pi) does not approach \frac{\pi}{2} exponentially fast on subsequences, |\mathrm{Re}(z^n)| should go to \infty. But how to prove that ? Here is a … Read more

On the values of ∏(p−1)/2k=1(e2πi/12−e2πik2/p)\prod_{k=1}^{(p-1)/2}(e^{2\pi i/12}-e^{2\pi i k^2/p}) for primes p>3p>3

In a recent preprint, I investigated Sp(x):=(p−1)/2∏k=1(x−e2πik2/p), where p is an odd prime and x is a root of unity. Motivated by Question 337879 and Question 338325, here I pose my conjecture on sp:=Sp(e2πi/12)=(p−1)/2∏k=1(e2πi/12−e2πik2/p) for primes p>3. Conjecture. Let p>3 be a prime. (i) If p\equiv13\pmod{24}, then s_p=i(-1)^{\frac{p-5}8+|\{1\le k<\frac p4:\ (\frac kp)=-1\}|}(x_p\sqrt3-y_p\sqrt p), where (\frac … Read more

Near Pochhammer symbols: the equation $(n)_m-(k)_l=2$ for integers greater than or equal to two

In this post I consider the following equation involving Pochhammer symbols, $$(n)_m-(k)_l=2\tag{1}$$ for positive integers $n\geq 2$ and $k\geq 2$, and positive integers $m\geq 2$ and $l\geq 2$. See if you need the definition of a Pochhammer symol from the artice Pochhammer Symbol of the encyclopedia Wolfram MathWorld. We denote the solutions as $(n,m;k,l)$, but … Read more

Finding integer solutions to n=a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−ac−bc)n=a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc)?

Is there anything known how to find integer solutions to n=a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−ac−bc) where n is a natural number ≠3,6mod9 and a,b,c∈Z? Notice that n=det where the matrix on the right hand side is the group matrix of the cyclic group C_3 as defined by Dedekind. Given a solution a,b,c in \mathbb{Q} to the equation above, n … Read more

On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$

My question is related to https://oeis.org/A269839. It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. Famous and simplest one is $6^3 = 3^3 + 4^3 + 5^3$ which Euler noted. I have also curiosity about solutions to $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge … Read more