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$p$-adic-valuation of an expression involving Bernoulli numbers
$begingroup$
Let $p = 43, 67$ or $163$ (three primes such that $h(mathbb{Q}(sqrt{-p})) = 1$) and consider $k = (p+1)/2$. I'm interested in computing the $p$-adic valuation of the expression
begin{equation}
1+frac{l}{B_l},
end{equation}
where $l = k + m(p-1),~mgeq 0$. Note that, by the Kummer congruences and the fact that $h(-p) equiv -2B_k~(mathrm{mod}~p)$, we have
begin{equation}
v_p(1+l/B_l)geq 1.
end{equation}
In fact, I have observed by numerical computations that this inequality is an equality for many value of $m$. I have also observed that if $m$ is such that $2m+1 = dp$ for an integer $d$, then the valuation of $1+l/B_l$ seems to be equal to $2$.
These observations seem very mysterious for me, so I was wondering if anybody had an idea to explain this? Thanks !
number-theory p-adic-number-theory bernoulli-numbers
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
$endgroup$
add a comment |
$begingroup$
Let $p = 43, 67$ or $163$ (three primes such that $h(mathbb{Q}(sqrt{-p})) = 1$) and consider $k = (p+1)/2$. I'm interested in computing the $p$-adic valuation of the expression
begin{equation}
1+frac{l}{B_l},
end{equation}
where $l = k + m(p-1),~mgeq 0$. Note that, by the Kummer congruences and the fact that $h(-p) equiv -2B_k~(mathrm{mod}~p)$, we have
begin{equation}
v_p(1+l/B_l)geq 1.
end{equation}
In fact, I have observed by numerical computations that this inequality is an equality for many value of $m$. I have also observed that if $m$ is such that $2m+1 = dp$ for an integer $d$, then the valuation of $1+l/B_l$ seems to be equal to $2$.
These observations seem very mysterious for me, so I was wondering if anybody had an idea to explain this? Thanks !
number-theory p-adic-number-theory bernoulli-numbers
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
$endgroup$
add a comment |
$begingroup$
Let $p = 43, 67$ or $163$ (three primes such that $h(mathbb{Q}(sqrt{-p})) = 1$) and consider $k = (p+1)/2$. I'm interested in computing the $p$-adic valuation of the expression
begin{equation}
1+frac{l}{B_l},
end{equation}
where $l = k + m(p-1),~mgeq 0$. Note that, by the Kummer congruences and the fact that $h(-p) equiv -2B_k~(mathrm{mod}~p)$, we have
begin{equation}
v_p(1+l/B_l)geq 1.
end{equation}
In fact, I have observed by numerical computations that this inequality is an equality for many value of $m$. I have also observed that if $m$ is such that $2m+1 = dp$ for an integer $d$, then the valuation of $1+l/B_l$ seems to be equal to $2$.
These observations seem very mysterious for me, so I was wondering if anybody had an idea to explain this? Thanks !
number-theory p-adic-number-theory bernoulli-numbers
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
$endgroup$
Let $p = 43, 67$ or $163$ (three primes such that $h(mathbb{Q}(sqrt{-p})) = 1$) and consider $k = (p+1)/2$. I'm interested in computing the $p$-adic valuation of the expression
begin{equation}
1+frac{l}{B_l},
end{equation}
where $l = k + m(p-1),~mgeq 0$. Note that, by the Kummer congruences and the fact that $h(-p) equiv -2B_k~(mathrm{mod}~p)$, we have
begin{equation}
v_p(1+l/B_l)geq 1.
end{equation}
In fact, I have observed by numerical computations that this inequality is an equality for many value of $m$. I have also observed that if $m$ is such that $2m+1 = dp$ for an integer $d$, then the valuation of $1+l/B_l$ seems to be equal to $2$.
These observations seem very mysterious for me, so I was wondering if anybody had an idea to explain this? Thanks !
number-theory p-adic-number-theory bernoulli-numbers
number-theory p-adic-number-theory bernoulli-numbers
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
edited Jan 30 at 17:16
David Ayotte
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
asked Jan 28 at 16:51
David AyotteDavid Ayotte
213
213
add a comment |
add a comment |
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