Bill's 'identity' is, of course, very clever. The right-hand-side of the 'identity' is in the algebraic closure of pi and exp(pi), whereas Yuri Nesterenko proved in 1996 that pi, exp(pi), and Gamma(1/4) are all algebraically independent of each other. My suspicion is that Bill deliberately chose a lattice generated by a large bunch of simple expressions in the algebraic closure of pi and exp(pi) and looked for small linear relations (using LLL, as Viet Elser mentioned) to engineer a 'counter-example' to Nesterenko's result. Best wishes, Adam P. Goucher
Sent: Tuesday, August 04, 2020 at 5:39 PM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] What do you mean, "Fake news"?
Out[364]= Gamma[1/4]==(1+√√5)^(3/2) √(5 (1+√2) (√2+√√5)) (1+√5)^(11/4) √√(3+√10) E^(-65π/3)π^(3/4) Sinh[20π]/2^(11/16)
In[372]:= N[%364,105]
Out[372]=3.62560990822190831193068515586767200299516768288006546743337799956991924353872912161836013672338430036147
==3.62560990822190831193068515586767200299516768288006546743337799956991924353872912161836013672338430036147 —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun