It should be p-adic differential equations and the Gross Koblitz formula On Tue, Jan 26, 2021 at 20:55 Victor Miller <victorsmiller@gmail.com> wrote:
The observation of Kummer eventually lead to the p-aficionado Gamma function of Morris, the https://en.wikipedia.org/wiki/Gross%E2%80%93Koblitz_formula?wprov=sfti1
And the theory of p-ADVI differential equations of Dwork.
On Tue, Jan 26, 2021 at 20:46 Dan Asimov <asimov@msri.org> wrote:
Thanks!
I wonder if there's a more concrete way to describe the power of 2.
I never understood my 2nd-grade teacher's explanation of borrowing for subtraction, so I invented my own way to subtract, which I use to this day.
(I think of it as an addition problem from the bottom up, and starting from the right, ask what the next digit of the difference has to be to make the addition work.)
—Dan
1.
Is this true for all n ≥ 0 ???
—Dan
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On Tuesday/26January/2021, at 5:32 PM, Victor Miller < victorsmiller@gmail.com> wrote:
This is an old result of Kummer. The number of powers of 2 dividing a binomial coefficient is the number of borrows when you write the top and bottom in base 2 and subtract.
On Tue, Jan 26, 2021 at 19:12 Dan Asimov <asimov@msri.org> wrote:
The OEIS sequence for
binomial(2n+1, n+1) = (2n+1)! / (n! (n+1)!)
(<https://oeis.org/A001700>) begins with
1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, 1352078, 5200300, 20058300, 77558760, 300540195, 1166803110, 4537567650, 17672631900, 68923264410, 269128937220, 1052049481860, 4116715363800, 16123801841550, 63205303218876, 247959266474052,
(0 ≤ n ≤ 25), for which
binomial(2n+1, n+1) = is odd if and only if n is of the form 2^K
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