Quicky: The product of palindromes is a palindrome, if there are no carries. In the 9*99*... case, if we allow m = -1 as a digit, the palindrome-product theorem needs a bit of fixup for nega-dromes like 1m*10m, but still seems to be relevant. Re JJs remark: There's a finite version of Euler's pentagonal number product. It's a special case of Thm 348 in Hardy & Wright's "Introduction to the Theory of Numbers", with a=-1: (1-x)(1-x^2)(1-x^3)...(1-x^j) = 1 - x (1-x^j)/(1-x) + x^3 (1-x^j)(1-x^(j-1))/(1-x)(1-x^2) - ... + (-1)^j x^(j(j+1)/2). The terms in the sum have x^triangle and the ratio of initial and final sections of the product. This ratio has a nice symmetry between the beginning and ending terms of the sum, a kind of power-series palindrome. We can plug in x = .1 or x = 10 for useful results. This should lead to regularities even in the middle digits of the product, until carries spoil the patterns. Q: If the quotient of two palindomes is an integer, must the quotient be a shoehorn-palindrome? SP is a palindrome with digits outside the normal range, with the carries propagated. 301 is an SP, with decimal representation 1*100 + 20*10 + 1, middle digit 20. It's tempting to do cancellations on the product-ratios from the sum, but there are some hazards: (1-x)(1-x^2)(1-x^3) is a divisor of (1-x^j)(1-x^(j-1))(1-x^(j-2)), both as integers and as polynomials, but if j = 7, the the ratio can't be split into individual polynomial terms (1-x^a)/(1-x^b) with b dividing a. Rich ------- Quoting Neil Sloane <njasloane@gmail.com>:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
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