Problems & Puzzles: Conjectures

Conjecture 10. Champions and Primorial Numbers

Conway & Odlyzko call the number (p_n-1 -p_n) a "champion for x" if it happens that it occurs most frequently for all the consecutive primes less than x. ; let us define C(x) as the symbol for that concept.

They conjecture also that C(x) only takes the following values : 2, 4, 6, 30, 210, 2310,… which means that C(x) is 4 or a primorial (p#=2*3*…*p). Is this true ?

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Mr. Marek Wolf has sent us the following e-mail comment:

": I am a co-author with Odlyzko and Rubinstein of the paper which practically  solves the champion problem. Even I have produced the table of approximate values of N^(n) at which the n-th champion 2x3x...xp_n wins, these numbers N^(n) grow very fast, roughly like n^n^n, see my web page: http://www.ift.uni.wroc.pl/~mwolf  

Or the following of Odlyzko and Rubinstein:

http://www.research.att.com/~amo/doc/recent.htm