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 mathbbZmathbbZ-like intervals II such that each xinIxinI has a standard factor. The proof is via Compactness, and the Chinese Remainder Theorem:



First, adjoin a constant symbol cc to our language. Let pipi be the ithith prime number, let qi=p2iqi=p2i, and let ri=p2i+1ri=p2i+1.



Define numbers aiai, bibi by recursion as follows:



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



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



Now, for each iinmathbbNiinmathbbN, let sigmaisigmai express "cc is congruent to aiai(modpipi)", let tauitaui express "cc is congruent to bibi(modpipi)," and let Sigma=lbracesigmai:iinmathbbNrbracecuplbracetaui:iinmathbbNrbraceSigma=lbracesigmai:iinmathbbNrbracecuplbracetaui:iinmathbbNrbrace. By the Chinese Remainder Theorem, every finite subset of SigmaSigma is consistent with True Arithmetic TATA, so by Compactness, SigmaSigma itself is consistent with TATA. So there is some nonstandard model of TATA in which SigmaSigma holds; clearly, in such a model, every number in the mathbbZmathbbZ-like interval centered on cc has a standard factor.



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

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