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Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:

(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?

(another member) Because it's false.

Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi_n(X)$ are $0,\pm1$. 0,\pm1$(Migotti, 1883). In other words, this holds true for$n=2^mp^kq^\ell$, where$p,q$are primes. On the other hand, it is false for$n=105=3\cdot5\cdot7$, because the coefficient of$X^7$(or of$X^{41}$as well) is$2$.-2$.

My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property?

2 spelling

Let me begin with what looks like a joke. According to a Bourbaki member, the following conservation conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:

(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?

(another member) Because it's false.

Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi_n(X)$ are $0,\pm1$. In other words, this holds true for $n=2^mp^kq^\ell$, where $p,q$ are primes. On the other hand, it is false for $n=105=3\cdot5\cdot7$, because the coefficient of $X^7$ (or of $X^{41}$ as well) is $2$.

My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property?

1

# Cyclotomic polynomials with coefficients $0,\pm1$

Let me begin with what looks like a joke. According to a Bourbaki member, the following conservation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:

(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?

(another member) Because it's false.

Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi_n(X)$ are $0,\pm1$. In other words, this holds true for $n=2^mp^kq^\ell$, where $p,q$ are primes. On the other hand, it is false for $n=105=3\cdot5\cdot7$, because the coefficient of $X^7$ (or of $X^{41}$ as well) is $2$.

My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property?