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Register a new userCubefree Trinomial Discriminants
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William Craig
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The discriminant of a polynomial of the form $pm x^n pm x^m pm 1$ has the form $n^n pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries that appear in the classification of these values of $n,m$. We prove that there are infinitely many pairs of integers $n,m$ for which this discriminant has no prime cube divisors. This result is extended to show that for infinitely many fixed $m$, there are infinitely many $n$ for which the discriminant has no prime cube divisor.
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In this paper, by using the tool of trinomial coefficients we study some determinant problems posed by Zhi-Wei Sun. For example, given any odd prime $p$ with $pequiv 2pmod 3$, we show that $2det[frac{1}{i^2-ij+j^2}]_{1le i,jle p-1}$ is a quadratic residue modulo $p$. This confirms a conjecture of Zhi-Wei Sun.
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