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}}