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Simon Plouffe

Publications and source records attributed to Simon Plouffe.

13 recordsLinked to original sources
arXiv2025

Verification of Gilbraith's conjecture up to 10$^{14}$

A calculation was performed to verify Proth-Gilbraith's conjecture for all prime numbers up to 0$^{14}$. The previous calculation was performed by Andrew Odlyzko in 1993 up to 0$^{13}$. This involves calculating the differences between consecutive primes in absolute value and starting over. The conjecture states that all lines except the first begin with 1. To prove it, it suffices to find a line beginning with 1 and followed only by 0 and 2.

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arXiv2025

Numbers in the base $e^\pi$

A large-scale experiment was conducted to find formulas relating to the base $e^\pi$. The numbers in this base are $$x = \sum_{n=0}^\infty {a(n)\over e^{\pi n}}$$ where $a(n)$ is taken from the OEIS catalog. These experiments were inspired by several facts. Indeed, it is known that the formula generating the partitions of integers is generated by an infinite product $$\prod_{k\ge1}^\infty {1\over 1-x^k} = \sum_{n=0}^\infty p(n)x^n$$ that when evaluated at $x=e^{-\pi}$ is equal to $${2^{3/8} \Gamma(3/4) \over \pi^{1/4} e^{\pi/24}}\ .\qquad\qquad (1)$$ By analyzing the 387500 sequences of the OEIS catalog, the model that was used is based on the fact that the infinite sum evaluated at $e^\pi$, is an expression that can be detected using a program like lindep from Pari-Gp. The process made it possible to find 793 expresssions similar to (1).

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arXiv2023

Pi and the primes

By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).

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arXiv2020

The calculation of p_n and pi_n

A new approach is presented for the calculation of p_n and pi_n which uses the Lambert W function. An approximation is first found and using a calculation technique it makes it possible to have an estimate of these two quantities more precise than those known from Cipolla and Riemann. The calculation of p(n) uses an approximation using the Lambert W function and an estimate based on a logarithmic least square curve (LLS) c_n.

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arXiv2019

A set of formulas for primes

A new set of formulas for primes is presented. These formulas are more efficient and grow much slower than the two known formulas of Mills and Wright. 3 new formulas are explained.

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arXiv2013

On the values of the functions zeta and gamma

An analysis of the zeta and gamma function is presented, using elementary functions like [] and {}, a general formula for the angle of zeta(1/2 + i*n) is found and the same for the gamma function.

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arXiv2011

Identities inspired by the Ramanujan Notebooks, second series

A series of formula is presented that are all inspired by the Ramanujan Notebooks [6]. One of them appears in the notebooks II about Zeta(3). That formula inspired others that appeared in 1998, 2006 and 2009 on the author's website and later in literature [1][2][3]. New formulas for π and the Catalan constant are presented and a surprising series of approximations. A new set of identities is given for Eisenstein series. All of the formulas are conjectural since they were found experimentally. A new method for the computation of p(n) is presented.

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arXiv2009

On the computation of the n^th decimal digit of various transcendental numbers

A method for computing the n'th decimal digit of pi in O(n^3 log(n)^3) time and with very little memory is presented here. The computation is based on the recently discovered Bailey-Borwein-Plouffe algorithm and the use of a new algorithm that simply splits an ordinary fraction into its components. The algorithm can be used to compute other numbers like zeta(3), pi*sqrt(3), pi^2 and 2/sqrt(5) ln(phi) where phi is the golden ratio. The computation can be achieved without having to compute the preceding digits. I claim that the algorithm has a more theoretical rather than practical interest, I have not found a faster algorithm or proved that it cannot exist.

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arXiv2009

A method for obtaining the algebraic generating function from a series

We describe here an experimental method that permits to compute a good candidate for the closed form of a generating function if we know the first few terms of a series. The method is based on integer relations algorithms and uses either two programs of symbolic computation: Maple or Pari-Gp. Some results are presented in the appendix. This method was tested on a set of sequences that were part of the incoming book on integer sequences (as of 1993). This method was presented at the FPSAC, Formal Power Series and Algebraic Combinatorics, Florence, June 1993.

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arXiv2009

Approximations of generating functions and a few conjectures

This is a collection of 1031 formulas that were generated by a computer program in 1992. The set is the database of integer sequences as of 1992 which contained 4568 sequences. These sequences were later published in the Encyclopedia of Integer Sequences by Neil J.A. Sloane and Simon Plouffe. The text explain the methodology used for finding a formula. The rest of the pages (over 500) is the collection of formulas found. NOTE : the sequences are numbered in the old system.

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arXiv2007

An efficient algorithm for the computation of Bernoulli numbers

This article gives a direct formula for the computation of B(n) using the asymptotic formula $$B (n) \approx 2 {\frac {n!}{π^{n}{2}^{n}}}$$ where n is even and $n >> 1$. This is simply based on the fact that $ζ(n)$ is very near 1 when n is large and since $B (n) = 2 {\frac {ζ(n) n!}{π^{n}{2}^{n}}}$ exactly. The formula chosen for the Zeta function is the one with prime numbers from the well-known Euler product for $ζ(n)$. This algorithm is far better than the recurrence formula for the Bernoulli numbers even if each B(n) is computed individually. The author could compute $B (750,000)$ in a few hours. The current record of computation is now (as of Feb. 2007) $B (5,000,000)$ a number of (the numerator) of 27332507 decimal digits is also based on that idea.

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