Testwiki:Reference desk/Archives/Mathematics/2020 October 30

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October 30

Roots of $X^{n-1}+1$ in $\mathbb{Z}/n$

Copied from WP:RD/MA/2020 October 22#Number of roots of polinomial in fields other than C.  --Lambiam 14:58, 30 October 2020 (UTC)

Experimentally, it appears that the polynomial $X^{n-1}+1,$ $n>1,$ has roots in $\mathbb{Z}/n$ iff its degree $n-1$ is odd. In the latter case we always have the root $X=n-1,$ and usually this is the only root. But sometimes there are more: for example, $X^{27}+1$ has 3 roots in $\mathbb{Z}/28$ ($X=3,$ $X=19,$ $X=27$), $X^{741}+1$ has 39 roots in $\mathbb{Z}/742,$ and $X^{945}+1$ has 105 roots in $\mathbb{Z}/946.$ I see no obvious pattern, but no doubt someone has studied this.  --Lambiam 08:05, 27 October 2020 (UTC)

If $p$ is an odd prime, the absence of roots of $X^{p-1}+1$ in $\mathbb{Z}/p$ follows from Fermat's little theorem. For composite odd moduli, their absence remains unexplained.  --Lambiam 17:50, 28 October 2020 (UTC)

The Chinese remainder theorem covers the case of products of distinct odd primes.--Jasper Deng (talk) 03:53, 29 October 2020 (UTC)

@JD — Can you give a proof sketch? I don't see the connection straightaway. I also remain curious about the odd distribution of the number of roots if the degree is odd.  --Lambiam 15:05, 30 October 2020 (UTC)

If n is even then (-1)^n-1 = -1, and so X^n-1+1=0 has a solution. The number of solutions is a bit trickier to determine. Note that since the map X→X^a is an endomorphism of the multiplicative group mod n, the number of solutions to X^a=b is equal to either 0 or the order of the kernel of this endomorphism. The n odd case seems more difficult, but the structure of the multiplicative group mod n is well known and this information can be utilized to derive a contradiction if X^n-1=-1 has a solution mod n. There may be a more elementary proof but I didn't see one. I haven't completely checked the details, but I think you can go a bit further: If a is even, n odd, a = 2^αb where b is odd, then X^a=-1 has a solution mod n only if each prime dividing n is congruent to 1 mod 2^α+1. --RDBury (talk) 22:48, 30 October 2020 (UTC)

PS. Template:Oeis gives the number of solutions of X^n-1=1 (mod n). If n is even this is the same number of solutions as X^n-1=-1. An easy(er) way to see this is that X^n-1=1 iff (-X)^n-1=-1. The first even number for which this value is >1 is 28, then 52, 66, 70, 76, ... --RDBury (talk) 07:49, 31 October 2020 (UTC)