I am wondering if it is possible to get a useful exact formula, or at least some useful asymptotics, for the partial sums of the Liouville function (OEIS sequence A002819) L(n)=n∑k=1λ(k), by exploiting the identity n∑k=1λ(k)⌊nk⌋=⌊√n⌋. The behavior of L(n) is strongly related to the Riemann hypothesis; see here for a great related MO question … Read more
Pick fourth power free integer n (p4 doesn’t divide n). Represent n as difference of possibly negative integer squares n=v2i−u2i. The goal is to find quadratic polynomial with integer coefficients (possibly square) f(x) maximizing the number of solutions to f(x)=ui. Solutions are integral points on the elliptic curve n=y2−f(x)2. Unless one can use the group … Read more
I posted this question to Mathematics Stack Exchange but got no answers. I hope that this question is advanced enough for this forum: In Pascal’s triangle, each number is the sum of the two numbers directly above it. I experimented with a modified Pascal’s triangle. First a n-tuple of natural numbers is put to the … Read more
Definition of CL for people who like number theory: Let m be a number with prime factorisation m=pn11…pnrr with ni>0. Define Im to be the incidence algebra of the divisor lattice of m. Up to isomorphism this just depence on the ni and not on the primes so lets define with L=[n1+1,…,nr+1], IL as the … Read more
From the literature, showed below, I know two gadgets that provide a way to know if a positive integer (a positive quantity of units) is composite or a prime number. I would like to know if in the literature or from your invention it is possible to show other different gadgets that provide us primality … Read more
In many areas of mathematics it is informative to conduct numerical experiments. But, it not uncommon that the searches do not lead to the examples or data one was hoping for. Since the numerical searches can be quite time consuming, it seems useful to share these negative results, so that others avoid spending time attempting … Read more
The computer found this, but couldn’t prove it. Let $\psi(n,x)$ denote the polygamma function. With precision 500 decimal digits we have: $$ \pi^2 = \frac{1}{4}(15 \psi(1, \frac13) – 3 \psi(1, \frac16)) $$ Is it true? In machine readable form: pi^2 == 1/4*(15*psi(1, 1/3) – 3*psi(1, 1/6)) Answer Note that $$ \psi(m,x) =(-1)^{m+1} m! \sum_{k=0}^{\infty} \frac{1}{(x+k)^{m+1}}. … Read more
The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows: Given an integer n>0 with binary expansion n=∑Ni=0di2i, make the substitution 1⟶10,0⟶0 in dNdN−1…d0, remove the final digit 0 and interpret the resulting … Read more
A while ago I threw the following at a numerical evaluator (in the present case I’m using wolfram alpha) ∏∞v=2v(v−1)√v≈3.5174872559023696493997936… Recently, for exploratory reasons only, I threw the following product at wolfram alpha ∏∞n=1n√1+1n≈3.5174872559023696493997936… (I have cut the numbers listed above off where the value calculated by wolfram alpha begins to differ) Are these products … Read more
the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for π that were discovered via the PSLQ algorithm as well as many other examples. however, it appears to glaringly leave out any description of the crucial step. it goes from discussing large accuracy floating point … Read more
