All Questions
Tagged with bernoulli-numbers nt.number-theory
9 questions with no upvoted or accepted answers
19
votes
0
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649
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Bernoulli & Betti numbers (of manifolds) and the prime 34511
The purpose of this question is to resolve a mystery surrounding the prime 34511 that has got me bogged down for a while now. If you only care about the number theory and not the motivation coming ...
- 11.9k
13
votes
0
answers
766
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A conjecture on p-divisibility of Bernoulli numbers
Is anyone aware of the history of the following conjecture on the $p$-divisibility of (the numerators) of Bernoulli numbers?
CONJECTURE: For $p$ an odd prime, and $k$ even with $2 \leq k \leq p-3$, $...
- 659
6
votes
0
answers
126
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divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1
For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...
- 61
3
votes
0
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186
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Combinatorial interpretation of Sylvester–Lipschitz formula?
If we denote the Bernoulli numbers by $B_n$, then
$$ {a^{k+1}(a^{2k} - 1) B_{2k} \over 2k}\in \mathbb{Z}$$
for any $a\in \mathbb{Z}$ and $k\in\mathbb{N}$. This fact is often called the Sylvester–...
- 82.7k
3
votes
0
answers
157
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Combinatorial interpretation for Möbius-poly-Bernoulli numbers
The Möbius-Bernoulli numbers ,are related to Dedekind Sums
$$\sum_{d|n}\frac{t\mu(d)}{e^{td}-1}=\sum_{k=0}^\infty M_k(n)\frac{t^k}{k!}$$ where $|t|<\frac{2\pi}{n}$, and $M_k(1)=B_k$.
We define the ...
user160903
3
votes
0
answers
237
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Values of Bernoulli polynomials at roots of unity
I am wondering if there are any nice results on the values of Bernoulli polynomials at roots of unity, besides those at 1 or -1.
- 31
2
votes
0
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84
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Question about infinitude of $m$-irregular primes
Let $p>2$ be a prime, $\zeta$ a primitive $p$th root of unity, and $m >0$ a square-free integer such that $m \not\equiv 3 \mod 4$ and $\gcd(m,p)=1$. Let $\chi$ be the imaginary quadratic ...
- 707
2
votes
0
answers
118
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Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$
For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...
- 1,836
2
votes
0
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212
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show that sequence $\{(-1)^n\Upsilon_n\}$ is convergent and strictly decreasing
Edit: Few years ago, I have posted my claim on $\Upsilon$ function regarding prime number but recently I have observed, last observation turns false that's way, (by putting $\Upsilon$ value in ...
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