[update (04/19/19): Vladica Andrejic, Alin Bostan and I extended the search up to 2^40 »]
[update (09/03/14): Vladica Andrejic and I wrote a paper regarding this problem and extended the search to 2^34 »]
Kurepa’s left factorial hypothesis states that , for all , where . This conjecture is also listed as a problem B44 in Guy’s “Unsolved problems in number theory”.
Official proof was published in 2004 by D. Barsky and B. Benzaghou “Nombres de Bell et somme de factorielles”. [update (12/12/12): This ‘proof’ turned out to be false »]
Vladica Andrejic (University of Belgrade) was suspicious that proof isn’t valid and conjectured that there are good chances that counterexample lies in the interval . Using the existing hardware, it isn’t possible to check all values from this interval, so I decided to extend the search beyond , which was covered in the latest attempt by P. Jobling.
Unfortunately, I didn’t find any counterexamples, but new values for were found.
Some previous searches:
– Miodrag Zivkovic, 1999,
– Yves Gallot, 2000,
– Paul Jobling, 2004,
My results for and
Several optimization tricks were used in order to reduce the number of required multiplications. This implementation required only ~1.8 multiplications per iteration. Also, a big advantage of modern CPUs is ability to multiply two 64bit registers without overflow. Four values of p were processed in the same loop in order to keep CPU-core busy. The program was written in asm/C, and ran at i7 quad-core CPU.