Re: Summation up to p-1 modulo p, p prime

the original question asked for what is: ? perhaps the difficuly is in what you meant by "it" in your post...your "it" and Pinkk's "it" may be 2 different things.

your orignal post (#11) was clearer.

i think this is an interesting problem, which "as-stated" doesn't have a definitive answer. i haven't looked very far into it, but my feeling is, that if p is a divisor of the "denominator" for a closed form expression of , each power will resolve to 1, and we get p-1. the "denominators" in question are the even-numbered terms in this sequence:

A064538 - OEIS

which are also the denominators of the even Bournoulli numbers, expressed in "lowest terms". it seems to me that this question is a "hard" question, which is belied by its simple statement.

the complete answer of this question, then, rests on proving the following:

let k be n even integer, and let denote the k-th Bernoulli number. if p divides the denominator of written in lowest terms, then

conjecture: the statement "p divides the denominator..." can be replaced by p-1 divides k.

any takers?

Re: Summation up to p-1 modulo p, p prime

From my previous post here we know that when .

If for some then by Fermat's Little Theorem. Otherwise, for which is not divisible by we get where is the remainder of upon division by . So again we obtain . In conclusion:

For , we have , and otherwise .

Re: Summation up to p-1 modulo p, p prime

Quote:

Originally Posted by

**Unbeatable0** From my previous post here we know that

when

.

If

for some

then

by Fermat's Little Theorem. Otherwise, for

which is not divisible by

we get

where

is the remainder of

upon division by

. So again we obtain

. In conclusion:

For

,

we have

, and otherwise

.

excellent! right, by FLT, each summand is congruent to 1 mod (p-1) (because p-1 is the order of the multiplicative group, and p-1 divides the power we're raising to). i should have thought of that myself. and every other case, we can reduce mod p to knock the exponents down to less than p-1.