Thursday, August 12, 2004

Number theory

I had a bit of fun last night coming up with descriptions of numbers, such as 'smallest fourth power [of integers] greater than unity', sixteen in other words. In mathematics these are known as 'exceptional numbers i.e. they have some property that is not shared with any other number. There's a nice little reductio ad absurdum proof that there's no smallest exceptional number: if there were, it would be exceptional.

But here's a real kicker. If we generously allow one to be a prime number, then I have seen it written that the modulus of all primes mod 24 is prime. Is this true? I've verified it in Mathematica* for the first million primes, i.e. from 2 to 15 485 863, but is it true for all primes? If so, twenty-four is a truly exceptional number.

*I dd this, talk about naive:

pt = Table[{Mod[Prime[i], 24], PrimeQ[Mod[Prime[i], 24]]}, {i, 1, 1000000}];

Select[pt, #[[2]] == False && #[[1]] != 1 &]

I had a bit of fun last night coming up with descriptions of numbers, such as 'smallest fourth power [of integers] greater than unity', sixteen in other words. In mathematics these are known as 'exceptional numbers i.e. they have some property that is not shared with any other number. There's a nice little reductio ad absurdum proof that there's no smallest exceptional number: if there were, it would be exceptional.

But here's a real kicker. If we generously allow one to be a prime number, then I have seen it written that the modulus of all primes mod 24 is prime. Is this true? I've verified it in Mathematica* for the first million primes, i.e. from 2 to 15 485 863, but is it true for all primes? If so, twenty-four is a truly exceptional number.

*I dd this, talk about naive:

pt = Table[{Mod[Prime[i], 24], PrimeQ[Mod[Prime[i], 24]]}, {i, 1, 1000000}];

Select[pt, #[[2]] == False && #[[1]] != 1 &]

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