I have nothing to contribute except the request that if you post the question to MathOverflow (and I think you should!), please post the link here so I can follow the discussion. Jim Propp On Wednesday, January 31, 2018, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
I've been looking at discriminants today.
Specifically, sequence https://oeis.org/A193681 Discriminant of minimal polynomial of 2*cos(Pi/n)
I've noticed that all these numbers are almost-powers of n.
For n with the following values 1, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179
The power is exactly 1, 1, 2, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 9, 22, 19, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 52, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88
What about the rest? Here are values {n,p,a,b} so that discriminant of minimal polynomial of 2*cos(Pi/n) is (n^p a^b)
discriminant( 2*cos(Pi/17) ) = 17^7 (Gauss's 17-gon) discriminant( 2*cos(Pi/168) ) = 168^48 189^(-8) discriminant( 2*cos(Pi/179) ) = 179^88
Is there any rhyme or reason to this sequence? --Ed Pegg Jr
{{1,1,1,1}, {2,1,2,-1}, {3,1,3,-1}, {4,1,2,1}, {5,1,1,1}, {6,1,2,1}, {7,2,1,1}, {8,4,2,-1}, {9,2,1,1}, {10,3,2,1}, {11,4,1,1}, {12,4,3,-2}, {13,5,1,1}, {14,5,2,1}, {15,3,3,-1}, {16,8,2,-1}, {17,7,1,1}, {18,4,12,1}, {19,8,1,1}, {20,6,2,4}, {21,5,3,-2}, {22,9,2,1}, {23,10,1,1}, {24,8,3,-4}, {25,8,5,1}, {26,11,2,1}, {27,7,3,1}, {28,10,2,4}, {29,13,1,1}, {30,4,20,2}, {31,14,1,1}, {32,16,2,-1}, {33,9,3,-4}, {34,15,2,1}, {35,10,5,-1}, {36,9,2,6}, {37,17,1,1}, {38,17,2,1}, {39,11,3,-5}, {40,16,5,-4}, {41,19,1,1}, {42,12,189,-2}, {43,20,1,1}, {44,18,2,4}, {45,9,1,1}, {46,21,2,1}, {47,22,1,1}, {48,16,3,-8}, {49,19,1,1}, {50,17,40,1}, {51,15,3,-7}, {52,22,2,4}, {53,25,1,1}, {54,15,2,3}, {55,18,5,-3}, {56,24,7,-4}, {57,17,3,-8}, {58,27,2,1}, {59,28,1,1}, {60,16,45,-4}, {61,29,1,1}, {62,29,2,1}, {63,15,3,-3}, {64,32,2,-1}, {65,22,5,-4}, {66,20,2673,-2}, {67,32,1,1}, {68,30,2,4}, {69,21,3,-10}, {70,18,56,2}, {71,34,1,1}, {72,18,2,18}, {73,35,1,1}, {74,35,2,1}, {75,18,32805,-1}, {76,34,2,4}, {77,27,7,-2}, {78,24,9477,-2}, {79,38,1,1}, {80,32,5,-8}, {81,23,3,2}, {82,39,2,1}, {83,40,1,1}, {84,24,189,-4}, {85,30,5,-6}, {86,41,2,1}, {87,27,3,-13}, {88,36,2,12}, {89,43,1,1}, {90,18,2,6}, {91,33,7,-3}, {92,42,2,4}, {93,29,3,-14}, {94,45,2,1}, {95,34,5,-7}, {96,32,3,-16}, {97,47,1,1}, {98,38,112,1}, {99,27,3,-9}, {100,35,2,10}, {101,49,1,1}, {102,32,111537,-2}, {103,50,1,1}, {104,44,2,12}, {105,20,405,-2}, {106,51,2,1}, {107,52,1,1}, {108,30,2,12}, {109,53,1,1}, {110,30,42592,2}, {111,35,3,-17}, {112,48,7,-8}, {113,55,1,1}, {114,36,373977,-2}, {115,42,5,-9}, {116,54,2,4}, {117,33,3,-12}, {118,57,2,1}, {119,45,7,-5}, {120,32,45,-8}, {121,52,1,1}, {122,59,2,1}, {123,39,3,-19}, {124,58,2,4}, {125,46,5,-1}, {126,27,56,3}, {127,62,1,1}, {128,64,2,-1}, {129,41,3,-20}, {130,36,1352,4}, {131,64,1,1}, {132,40,2673,-4}, {133,51,7,-6}, {134,65,2,1}, {135,30,5,-3}, {136,60,2,12}, {137,67,1,1}, {138,44,4074381,-2}, {139,68,1,1}, {140,36,448,4}, {141,45,3,-22}, {142,69,2,1}, {143,55,11,-1}, {144,48,3,-24}, {145,54,5,-12}, {146,71,2,1}, {147,39,2711943423,-1}, {148,70,2,4}, {149,73,1,1}, {150,40,45,-10}, {151,74,1,1}, {152,68,2,12}, {153,45,3,-18}, {154,50,3872,2}, {155,58,5,-13}, {156,48,9477,-4}, {157,77,1,1}, {158,77,2,1}, {159,51,3,-25}, {160,64,5,-16}, {161,63,7,-8}, {162,47,384,1}, {163,80,1,1}, {164,78,2,4}, {165,36,820125,-2}, {166,81,2,1}, {167,82,1,1}, {168,48,189,-8}, {169,74,13,1}, {170,64,10625,-4}, {171,51,3,-21}, {172,82,2,4}, {173,85,1,1}, {174,56,138706101,-2}, {175,53,1715,-1}, {176,80,11,-8}, {177,57,3,-28}, {178,87,2,1}, {179,88,1,1}, {180,36,2,24}} _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun