If I were your doctor I would prescribe Mozart. ---------- Forwarded message --------- From: Julian Ziegler Hunts <julianj.zh@gmail.com> Date: Sat, Nov 17, 2018 at 8:47 PM Subject: Re: [math-fun] Rational-coefficient polynomial, applied to integers To: Bill Gosper <billgosper@gmail.com> Two more obvious errors in (v): that should be a product, not a sum, and there's a fencepost error. The actual formula is product(p^(a(j)(d-jp+1)), j=1,…,l-1)*p^((m-h)(d-lp+1)), which can probably be rearranged to something nicer. Apart from that, I think everything is correct. The faster method I mentioned is the following: if m≤p, then a polynomial which is identically zero mod p^m is a sum of multiples of (x^p-x)^m, p*(x^p-x)^(m-1), …, p^(m-1)*(x^p-x), and this condition can, I think, be checked faster than plugging in (it definitely can for m very small, e.g. m=3). Julian On Sat, Nov 17, 2018 at 1:01 AM Bill Gosper <billgosper@gmail.com> wrote:

Man, before falling asleep (like now) I can't even stay focused enough to make your edits, let alone find bugs. Instead I wake up to find characters repeated thousands of times. —Bill

On Sat, Nov 17, 2018 at 12:14 AM Julian Ziegler Hunts < julianj.zh@gmail.com> wrote:

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Naturally I found an error while struggling to fall asleep: the sum in (v) is the number of such polynomials of degree at most d.

I think the (somewhat) simpler/faster methods in (iv) apply up to p^p, but fail for p^(p+1), though they may be able to be modified.

Julian

On Fri, Nov 16, 2018 at 11:47 PM Julian Ziegler Hunts < julianj.zh@gmail.com> wrote:

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You can forward it if you want to. It's not like everything posted to math-fun is correct, and if someone else finds an error or refinement before I do, that's fine with me.

Julian

On Fri, Nov 16, 2018 at 11:40 PM Bill Gosper <billgosper@gmail.com> wrote:

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Do you want me to hold off forwarding this to math-fun? —Bill