🚀 go-pugleaf

>>model of the integers, in which there would be two (nonstandard) integers,
>>n and n+2, both of which would be divisible by all standard integers. 

>It's a reasonable question (to which I don't know the answer) what the
>result of Kemeny is that wft is misremembering.

Sorreeee; bit of a boo-boo here. Brain damage my only excuse.

Should be:  there is a nonstandard integer n, which is divisible by all
standard integers. Thus  (n+1) and (n-1) are not divisible by any of them,
and are thus "standardly" twin primes. Of course, they may still be
composites in the whole nonstandard system. So this doesn't really help out
on the twin prime conjecture.

However; as I recall, Kemeny observed that (n-1), (n+1) are the *only* possible
primes in the "row" of numbers   .... , n-2 , n-1 , n , n+1 , n+2 , n+3 , ... 
Hence there are no 3-tuples of primes of type (p, p+2 , p+4) in the nonstandard
part of the model, (or any n-tuples with n>2). 

Thus, the n-tuple-primes conjectures of Hardy would all be false. The only
problem with this whole operation is showing that there can be a model with
*all* the "rows" having such an `n' in them. Kemeny reluctantly left this 
loop-hole unplugged.

Has anyone heard of the paper ? Or more precisely, followups to it ?

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            Bill Taylor              wft@math.canterbury.ac.nz 
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 Stop press ! I have just scaveged up a copy of Kemeny's paper.

It is "Undecidable problems of elementary number theory",
       Math Annalen, Bd. 135, (1958), pp 160-169.

I would still appreciate hearing of any follow-ups, though !