[seqfan] Maple packages numbertheory replaced by NumberTheory

[Lang, Wolfdieter (ITP)]

Dear Seqfans,

For those Maplers not already aware of this (as for me up to yesterday).
Thomas Richard from Maplesoft Support asked me to spread the following  information.

With Maple 2016 the successor of the package numtheory is the improved package NumberTheory. With Maple 2019 the package numtheory is declared 'depreciated' and is no longer supported.


I contacted Maple because in numtheory they have legendre(odd integer, 2) and jacobi(odd integer,2) both as -1, which is not nice, or even wrong. Usually Legendre(n, prime) is defined only for odd primes (similarly Jacobi(n, prime) = Legendre(n, prime)) and also gcd(n, prime) = 1. If quadratic residues modulo a prime are at all defined for prime = 2, then, if one also wants to use Legendre(odd number, 2) at all, this should then be taken as +1. But I prefer not to use the trivial case prime = 2 for these symbols used for theorems like Euler's criterion or reciprocity.

Now LegendreSymbol(5,2) gives in Maple an error with some explanation.


An example in OEIS where this came up, by a remark of Michel Marcus, is A038893 (see the pink box discussion in history).


What about Mathematica's  JacobiSymbol[n, m=2]? WolframAlpha gives -1, This is not covered by the definition in

http://functions.wolfram.com/NumberTheoryFunctions/JacobiSymbol/02/ where only nonnegative odd m is used.

Regards.

Wolfdieter Lang