This paper investigates some algebraic properties of the chromatic polynomials of theta graphs, i.e. graphs which have three internally disjoint paths sharing the same two distinct end vertices. Weâ€¦ (More)

If qÎ± is the Euler factor of an odd perfect number N, then we prove that its so-called index Ïƒ(N/qÎ±) ï¿½ qÎ± â‰¥ 32 Ã— 5Ã— 7 = 315. It follows that for any odd perfect number, the ratio of the non-Eulerâ€¦ (More)

Let E/Q be a modular elliptic curve, and p > 3 a good ordinary or semistable prime. Under mild hypotheses, we prove an exact formula for the Î¼-invariant associated to the weight-deformation of theâ€¦ (More)

We prove the exceptional zero conjecture is true for semistable elliptic curves E/Q over number fields of the form F ( e2Ï€i/q n ,âˆ† 1/q 1 , . . . ,âˆ† 1/q d ) where F is a totally real field, and theâ€¦ (More)