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Here is the proof of Gersonides [Levi ben Gershon] (1343) for $2^n-3^m=1$. It uses nothing more that arithmetic modulo $8$. Case I: $m$ is even. Then $3^m$ is 1 mod 4, so $2^n$ is 2 mod 4, implying $ …

"He said (and I never understood this comment so feel free to fill me in) that S_k(1;Q) being irreducible as a Hecke module was related to (equivalent to?) a certain L-value not vanishing, and L-value …

If my understanding is correct, for "squarefree part" can be "squarefree kernel" in other cases, the generating Dirichlet series is $${\zeta(2s)\zeta(s-1)\over\zeta(2s-2)}=\prod_p\biggl(1+{p\over p^s} …

For this context, though not so highbrow: Wagstaff, Prime Numbers with a Fixed Number of One Bits or Zero Bits in Their Binary Representation, Experiment Math 10 (2001), 267-273. eudml link, http://ww …

"One could try to estimate the size of the Tate-Shafarevich group of a "random" elliptic curve, to get an idea of how often local solvability implies global solvability, but even if one does this it i …

I will copy a comment of mine about Jehanne over here. His paper is: http://dx.doi.org/10.1006/jnth.2001.2656 Jehanne has 4 totally real Examples 2-5 (page 353-6), including the one of Booker. He al …

Hmm, a problem. In 4 hours Magma can compute this: > f:=x^24 - 50726*x^22 + 152929135*x^20 - 158664037068*x^18 + 63787035668165*x^16 - 9040633522810414*x^14 + 287094814384960835*x^12 - 205003861 …

Actually, I should read the help with Magma, for the direction to ArtinRepresentations is there. f:=14488688572801 - 2922378139308818*x^2 + 134981448876235615*x^4 - 1381768039105642956*x^6 + 4 …

I got it to work! First the preliminary data: _<x> := PolynomialRing(Rationals()); f5 :=344 + 3106*x - 1795*x^2 - 780*x^3 - x^4 + x^5; …

Here is a trial proof for the question over $Q_p$. Write $J[f(x)^k]$ for the general Jordan form of a irreducible $f$, being $k$ identical blocks joined by 1's in general (minimal polynomial of block …

Here is Magma code that gets you the answer in a few seconds. I made a special case for the bad primes, and did them by hand. _<x> := PolynomialRing(Rationals()); f5 := 344 + 3106*x - 1795*x^2 - 780* …

Back in 2007 or so, at a tea I heard a noted expert in the field pooh-poohing it (for instance, sign errors in the Stokes analogue), and he seemed not to want to read any more re-hashes (he had seen m …

"I expect a classification for general g to be longer, but maybe its managable when g = 2?" The Experimental Mathematics paper of Dieulefait for g=2 uses this. He quotes Mitchell from 1914. http://w …

"At the level of L-functions, this should force the L-fn attached to the weight two form of g to vanish to order the order of vanishing of the L-fn of the weight 4 form f. How could two modular forms …

Magma is not facile here but works, but maybe SAGE can do the same. You get $K(j,E[3])/K$ to be a degree 12 and cyclic Galois group, for the $E$ I think you want. > jrel:=PowerRelation(jInvariant((1+ …