Tuesday, 12 June 2007

nt.number theory - Composite pairs of the form n!-1 and n!+1

As far as nonstandard models go: we can indeed get mathbbZ-like intervals I such that each xinI has a standard factor. The proof is via Compactness, and the Chinese Remainder Theorem:



First, adjoin a constant symbol c to our language. Let pi be the ith prime number, let qi=p2i, and let ri=p2i+1.



Define numbers ai, bi by recursion as follows:



a0=0, an+1=minlbracex:forallkinmathbbN,jlen(cnot=aj+kqj)rbrace



b0=0, bn+1=minlbracex:forallkinmathbbN,jlen(cnot=bj+krj)rbrace



Now, for each iinmathbbN, let sigmai express "c is congruent to ai(modpi)", let taui express "c is congruent to bi(modpi)," and let Sigma=lbracesigmai:iinmathbbNrbracecuplbracetaui:iinmathbbNrbrace. By the Chinese Remainder Theorem, every finite subset of Sigma is consistent with True Arithmetic TA, so by Compactness, Sigma itself is consistent with TA. So there is some nonstandard model of TA in which Sigma holds; clearly, in such a model, every number in the mathbbZ-like interval centered on c has a standard factor.



I have no idea whether every nonstandard model has such an interval, however.

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