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did u know that e^(pi*sqrt(163)) is almost an integer??? it is very close to 262537412640768744=640320^3+744
the reason for this is that the j-invariant j(t) is an algebraic integer for all quadratic irrationals t in the upper half plane, and its degree is the class number of the field Q(t) formed by adjoining t to the rational numbers (which is to say, how many "classes" of ideal there are, or more intuitively how far Q(t) strays from unique factorization). the set of quadratic irrationals with real part 1/2 that give unique factorization is well known to be (1+sqrt(-d))/2 where d is one of 9 special integers called heegner numbers, of which 163 is the largest
if we set q=e^(2pi*i*t), then we get the well known power series expansion j(t)=1/q+744+196884q+21493760q^2+... and we note that for t=(1+i*sqrt(163))/2 all terms except for the first two are small enough to be neglected. given that j(t) for this value is -640320^3, we arrive at the desired (almost-)equation
what are ur favrite mathematical results? tell me in tha comments ^w^
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L. Eleanor "Ellie" Nguyen
ooh Ramanujan's constant
Ah also my favorite mathematical results (comments after ### because I'm a programmer)
1. X := (X -> Y)
2. X -> X ### identity
3. X -> (X -> Y) ### substitute right with 1
4. X -> Y ### contraction
5. X
6. Y
Curry's Paradox, in natural language, set theory, and various types of logic. Y is provable by the existence of a sentence X stating if X, then Y. A logic with these rules allows for the provability of all things, especially in naive logics and set theory. It could point to a general structure of the nature of logical paradoxes as a whole, given that it exists without a negation (a more famous paradox, Russell's paradox, involves negation -- set of sets not containing itself).
It also has a fun connection to Lob's theorem.
by L. Eleanor "Ellie" Nguyen; ; Report
ahhhh an actual logic bomb
thank u haskell curry very cool !!!
by bethany :3; ; Report