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Those of you deterred by Macsyma's ugly ASCII displays of my recent eta evaluations are invited to browse http://gosper.org/etavals.html . It's already obsolete, and will likely get updated.
E.g., with a bunch of identities as follow: For positive integers a,b,c, let p{a,b,c} := the lowest degree (trivariate) polynomial equation satisfied by eta(q^a),eta(q^b), and eta(q^c). E.g., 8 16 16 8 24 p{1,2,4} = (16 h1 h4 + h1 h4 = h2 ), 3 9 6 6 9 3 12 p{1,3,9} = (27 h1 h9 + 9 h1 h9 + h1 h9 = h3 ), where h2 := eta(q^2), etc. (Maybe someone can tell us why these relations always seem to exist.) If q -> q^d, we have p{a,b,c} <=> p{da,db,dc} and can thus scale by common divisors. With resultants, we can also combine p{a,b,c} p{a,b,d} = p{b,c,d}. E.g., p{1,2,4} p{2,4,8} = p{1,2,8} 24 8 32 24 16 24 24 24 16 = (1048576 h1 h2 h8 + 131072 h1 h2 h8 + 4864 h1 h2 h8 48 16 24 32 8 48 8 8 64 + 16 h1 h8 + 48 h1 h2 h8 + h1 h2 h8 = h2 ), and similarly for p{1,4,8}. So now we can get p{2^a,2^b,2^c} and p{3^a,3^b,3^c}. But row reduction on series can, *in principle*, find *any* p{a,b,c}, e.g., 6 8 12 12 8 6 24 2 p{1,2,5} = (- 3125 h1 h2 h5 - 250 h1 h2 h5 + 256 h2 h5 24 2 18 8 + h1 h5 = h1 h2 ).
From this you may enjoy figuring out how to get p{1,5,25}. Also notice that you can't get p{1,2,5} from p{1,2,4} and p{1,5,25}. So now we have p{2^a 5^b, 2^c 5^d, 2^e 5^f}. But I said we have p{a,b,c} from row reduction, no? *In practice*, row reduction is more strenuous than resultants, and seems limited to exponents of 50 or so, bad news because it's the only source of new primes. (Well, there's always numeric lattice reduction.) But resultants are strenuous, too! A couple of days ago Mathematica disgorged a resultant for p{4,6,9}, a geometric progression with a presumably simple polynomial, but got > 2000 terms, with thousand-digit coefficients, and square-free! Needless to say, the factorization is still in progress.
The Jacobi imaginary transformation provides one more operation, the "flip" involution: /p{a,b,c} = p{bc,ca,ab}, which is at least a nonstrenuous shortcut. The shapes of these polynomials are surprising. p{1,2,7} (deg 76) has 12 terms; p{1,3,7} (deg 26) has only 8. p{1,P,P^2} has only one P term for P=2,3,and 5, but 8 3 4 2 2 4 3 4 p{1,7,49} = (h7 - 49 h1 h49 h7 - 35 h1 h49 h7 - 7 h1 h49 h7 7 2 6 3 5 4 4 - 343 h1 h49 - 343 h1 h49 - 147 h1 h49 - 49 h1 h49 5 3 6 2 7 - 21 h1 h49 - 7 h1 h49 = h1 h49). p{1,5,6} has 247 terms, starting with 6 36 288 267901468048997825931274052676501085034012492775816057012130948209391921577066496 h1 h5 h6 . So far I only have primes through 11, which is not to say p{49,77,121} ! However, Michael Somos mentions in a paper at somos@harary.math.georgetown.edu "my database of thousands of eta-product identities which I update frequently and is available upon request", so he might have lots of primes. OtOH, his paper is about relating >>3 etas, which may also be true of much of his collection. --rwg