Amanda Folsom and Ken Ono’s Error Could Have (and Should Have!) been Discovered by Their Computers (in Less than Three Seconds!) Shalosh B. EKHAD ^{1}

Experimental Mathematics giants, David Bailey and Jonathan Borwein, describe ([BB]) how the use of computers is revolutionizing doing mathematics. The most obvious and trivial use, that even “pure” mathematicians (like Amanda Folsom and Ken Ono) should be able to do, is the empirical checking of formulas and statements. Unfortunately, they don’t. Frank Garvan (see [AGL]) discovered that the statement of Theorem 1.2 in [FO] is false as stated. You can’t trust humans, they are such screw-ups! This prompted me to find out for myself. Once the short Maple code (written in a few minutes by D. Zeilberger):

http://www.math.rutgers.edu/~zeilberg/tokhniot/KenAmandaError ,

is downloaded and read (or copied-and-pasted) into a Maple session, typing:

CheckKenAmanda(507); ,

returns (in less than three seconds!):

  false .

(Note that 24 · 507 − 1 = 23 · (23)^{2}, but spt(507) is even). The fascinating paper [AGL] contains a corrected statement of this theorm, and much nicer proofs (not using Maass forms, but rather elegant combinatorial arguments). It also describes how to fix the original proof of [F0]. References

[AGL] G.E. Andrews, F. G. Garvan, and J. Liang, Self-conjugate vector partitions and the parity

of the spt-function, http://www.math.ufl.edu/~fgarvan/papers/spt-parity.pdf .

[BB] David H. Bailey and Jonathan M. Borwein, Exploratory Experimentation and Computation, Notices of the Amer. Math. Soc. 58 (2011), 1410-1419.

[FO] A. Folsom and K. Ono, The spt-function of Andrews, Proc. Natl. Acad. Sci. USA 105 (2008), 20152-20156.

1 c/o zeilberg at math dot rutgers dot edu.

Exclusively published in: http://www.math.rutgers.edu/~zeilberg/pj.html . Nov. 22, 2011.