The table is Salem polynomial. The first is the Lehmer polynomial. 1 + x - x^3 - x^4 - x^5 - x^6 - x^7 + x^9 + x^10 If you look at the discriminants of these polynomials and take the square roots, you'll get the following series. {36497, 53995179961, 41783473, 34554953, 38569, 110103089329, 24217 Sqrt[5], 1426013608589 Sqrt[11], 204970773122077, 1823954309 Sqrt[3]} This corresponds to 1, 1, 1, 1, 1, 1, 5, 11, 1, 3 Continuing gives the sequence I mentioned. {1, 1, 1, 1, 1, 1, 5, 11, 1, 3, 1, 7, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 13, 19, 5, 1, 1, 13, 5, 1, 7, 1, 1, 41, 1, 1, 3, 19, 1, 1, 1, 1, 5, 3, 7, 1} --Ed Pegg Jr On Thu, Sep 13, 2018 at 1:10 PM Neil Sloane <njasloane@gmail.com> wrote:
Ed, As a collector of sequences, naturally I was interested in those 47 small numbers - the list is new to me. The link you gave shows a table which begins like this:
1.) 10 1.176280818259917506544070338474 1 1 0 -1 -1 -1 2.) 18 1.188368147508223588142960958629 1 -1 1 -1 0 0 -1 1 -1 1 3.) 14 1.200026523987391518902962100414 1 0 0 -1 -1 0 0 1 4.) 14 1.202616743688604261118295415948 1 0 -1 0 0 0 0 -1 5.) 10 1.216391661138265091626806311199 1 0 0 0 -1 -1 6.) 18 1.219720859040311844169606760414 1 -1 0 0 0 0 0 0 -1 1 7.) 10 1.230391434407224702790177938975 1 0 0 -1 0 -1 8.) 20 1.232613548593121003962731694807 1 -1 0 0 0 -1 1 0 0 -1 1 9.) 22 1.235664580389747308105169351531 1 0 -1 -1 0 0 0 1 1 0 -1 -1
How did you get {1, 1, 1, 1, 1, 1, 5, 11, 1, 3, 1, 7, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 13, 19, 5, 1, 1, 13, 5, 1, 7, 1, 1, 41, 1, 1, 3, 19, 1, 1, 1, 1, 5, 3, 7, 1} ?
If you take the discriminant of the polynomial and divide by the corresponding prime the result is a perfect square number.
What are the corresponding primes? 10/prime != a square ! 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 Thu, Sep 13, 2018 at 11:27 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
See eg. https://en.wikipedia.org/wiki/Salem_number --- the short monograph cited there Salem, R. (1963) "Algebraic numbers and Fourier analysis" is full of fascinating stuff which (AFAIK) is available nowhere else.
WFL
On 9/13/18, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
At http://www.cecm.sfu.ca/~mjm/Lehmer/lists/SalemList.html is a list of the 47 smallest known Salem number polynomial.
Here's a list of 47 small numbers. {1, 1, 1, 1, 1, 1, 5, 11, 1, 3, 1, 7, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 13, 19, 5, 1, 1, 13, 5, 1, 7, 1, 1, 41, 1, 1, 3, 19, 1, 1, 1, 1, 5, 3, 7, 1}
If you take the discriminant of the polynomial and divide by the corresponding prime the result is a perfect square number.
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun