2014-11-28, 14:51 | #1 |
Nov 2011
2^{2}×3×23 Posts |
Finding all divisors kn + 1 of P(n) for various polynomials P
Let be a monic polynomial with such that . I was interested in looking at the divisors of the values P(n) of the form kn+1. It appears that, given one divisor
there is an infinite series of the divisors of this form which is given by the equations For the polynomial 1 one can classify all such series. They are "generated" by the pairs for an arbitrary . This in turn implies (with some efforts to be made) that numbers do not have (non-algebraic) divisors of the form . Concerning the polynomial we also have series of divisors of P(n) generated by the pairs . Additionally the pairs generate infinite series of divisors of P(n). However a basic search among small numbers shows that there are still "exceptional" pairs which generate the divisors , the smallest of them is (3,11). With some efforts one can check that all (non-algebraic) divisors of numbers must come from an "exceptional" pair . I do not know how to classify the "exceptional" pairs . I conducted a search for all pairs with and additionally with . In total there are 201 different infinite series found. Also it seems that they are more less equidistributed on a coordinate plane. It would be very interesting to find a way to classify all of the exceptional pairs . In particular it may give us all divisors of numbers . |
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