******************************************* * * Length 3 Theorems for bases * {4, 16, 18, 24, 55, 81} * ******************************************* For a given base b and number of digits l, let N(l,b) denote the number of limiting orbits occuring over all possible input numbers num with l digits in base b to the iterative procedure listPhenom(num,b), and let L(l,b) denote the list of lengths of each orbit, sorted in increasing order. ************************************************* ************************************************* Odd-Base Conjecture(s) ************************************************* ************************************************* For an odd b >= 3, let n be the unique integer s.t. b = 2*n+1. Then we have the following theorem: N(3,b) = 1. Specifically, L(3,b) = [2]. In particular, the single limiting orbit is given by [[n-1, 2*n, n+1], [n, 2*n, n]]. ***************** base b = 55 ***************** For b = 55, n = 27. The list of all limiting orbits is: [[26, 54, 28], [27, 54, 27]] Thus N(3,b) = 1 and L(3,b) = [2]. Further, the limiting orbit is [[26, 54, 28], [27, 54, 27]]. Thus the conjecture is true. ***************** base b = 81 ***************** The base b = 81 has not been pre-computed, but since in this case n = 40: the theorem says that N(3,b) = 1 and L(3,b) = [2]. Further, it says that the limiting orbit is [[39, 80, 41], [40, 80, 40]]. ************************************************* ************************************************* Even-Base Conjecture(s) ************************************************* ************************************************* For an even b >= 2, let n be the unique integer s.t. b = 2*n Then we have the following theorem: Theorem: N(3,b) = 1. Specifically, L(3,b) = [1]. In particular, the single limiting orbit is given by [[n-1, 2*n-1, n]]. ***************** base b = 4 ***************** For b = 4, n = 2. The list of all limiting orbits is: [[1, 3, 2]] Thus N(3,b) = 1 and L(3,b) = [1]. Further, the limiting orbit is [[1, 3, 2]]. Thus the conjecture is true. ***************** base b = 16 ***************** For b = 16, n = 8. The list of all limiting orbits is: [[7, 15, 8]] Thus N(3,b) = 1 and L(3,b) = [1]. Further, the limiting orbit is [[7, 15, 8]]. Thus the conjecture is true. ***************** base b = 18 ***************** For b = 18, n = 9. The list of all limiting orbits is: [[8, 17, 9]] Thus N(3,b) = 1 and L(3,b) = [1]. Further, the limiting orbit is [[8, 17, 9]]. Thus the conjecture is true. ***************** base b = 24 ***************** For b = 24, n = 12. The list of all limiting orbits is: [[11, 23, 12]] Thus N(3,b) = 1 and L(3,b) = [1]. Further, the limiting orbit is [[11, 23, 12]]. Thus the conjecture is true. ******************************************* * * Length 4 conjecture(s) for bases * {4, 16, 18, 24, 55, 81} * ******************************************* For a given base b and number of digits l, let N(l,b) denote the number of limiting orbits occuring over all possible input numbers num with l digits in base b to the iterative procedure listPhenom(num,b), and let L(l,b) denote the list of lengths of each orbit, sorted in increasing order. ************************************************* ************************************************* Odd-Base Conjecture(s) ************************************************* ************************************************* For an odd b >= 5, let n be the unique integer s.t. b = 2*n+1. Finally, let U be the union of all the 4-digit numbers which occur in some limiting orbit. Then we have the following conjectures: Conjecture A (Parity Conjecture): If n mod 4 = 0 or 1 then N(4,b) is even, and if n mod 4 = 2 or 3, then N(4,b) is odd. Conjecture B (Orbit Lengths Division Conjecture): Each integer in L(4,b) divides m. Conjecture C (Elements Conjecture): U = {seq(seq([2*k+1, 2*i, 2*(n-i)-1, 2*(n-k)], i=0..k-1), k=1..n-1)}. ***************** base b = 55 ***************** The base b = 55 has not been pre-computed, but since in this case n = 27: n mod 4 = 3, and so Conjecture A says that N(4,b) is odd.Further, Conjecture C says that U = {seq(seq([2*k+1, 2*i, 53-2*i, 54-2*k], i=0..k-1), k=1..26)}, which more explicitly is: {[3, 0, 53, 52], [5, 0, 53, 50], [5, 2, 51, 50], [7, 0, 53, 48], [7, 2, 51, 48], [7, 4, 49, 48], [9, 0, 53, 46], [9, 2, 51, 46], [9, 4, 49, 46], [9, 6, 47, 46], [11, 0, 53, 44], [11, 2, 51, 44], [11, 4, 49, 44], [11, 6, 47, 44], [11, 8, 45, 44], [13, 0, 53, 42], [13, 2, 51, 42], [13, 4, 49, 42], [13, 6, 47, 42], [13, 8, 45, 42], [13, 10, 43, 42], [15, 0, 53, 40], [15, 2, 51, 40], [15, 4, 49, 40], [15, 6, 47, 40], [15, 8, 45, 40], [15, 10, 43, 40], [15, 12, 41, 40], [17, 0, 53, 38], [17, 2, 51, 38], [17, 4, 49, 38], [17, 6, 47, 38], [17, 8, 45, 38], [17, 10, 43, 38], [17, 12, 41, 38], [17, 14, 39, 38], [19, 0, 53, 36], [19, 2, 51, 36], [19, 4, 49, 36], [19, 6, 47, 36], [19, 8, 45, 36], [19, 10, 43, 36], [19, 12, 41, 36], [19, 14, 39, 36], [19, 16, 37, 36], [21, 0, 53, 34], [21, 2, 51, 34], [21, 4, 49, 34], [21, 6, 47, 34], [21, 8, 45, 34], [21, 10, 43, 34], [21, 12, 41, 34], [21, 14, 39, 34], [21, 16, 37, 34], [21, 18, 35, 34], [23, 0, 53, 32], [23, 2, 51, 32], [23, 4, 49, 32], [23, 6, 47, 32], [23, 8, 45, 32], [23, 10, 43, 32], [23, 12, 41, 32], [23, 14, 39, 32], [23, 16, 37, 32], [23, 18, 35, 32], [23, 20, 33, 32], [25, 0, 53, 30], [25, 2, 51, 30], [25, 4, 49, 30], [25, 6, 47, 30], [25, 8, 45, 30], [25, 10, 43, 30], [25, 12, 41, 30], [25, 14, 39, 30], [25, 16, 37, 30], [25, 18, 35, 30], [25, 20, 33, 30], [25, 22, 31, 30], [27, 0, 53, 28], [27, 2, 51, 28], [27, 4, 49, 28], [27, 6, 47, 28], [27, 8, 45, 28], [27, 10, 43, 28], [27, 12, 41, 28], [27, 14, 39, 28], [27, 16, 37, 28], [27, 18, 35, 28], [27, 20, 33, 28], [27, 22, 31, 28], [27, 24, 29, 28], [29, 0, 53, 26], [29, 2, 51, 26], [29, 4, 49, 26], [29, 6, 47, 26], [29, 8, 45, 26], [29, 10, 43, 26], [29, 12, 41, 26], [29, 14, 39, 26], [29, 16, 37, 26], [29, 18, 35, 26], [29, 20, 33, 26], [29, 22, 31, 26], [29, 24, 29, 26], [29, 26, 27, 26], [31, 0, 53, 24], [31, 2, 51, 24], [31, 4, 49, 24], [31, 6, 47, 24], [31, 8, 45, 24], [31, 10, 43, 24], [31, 12, 41, 24], [31, 14, 39, 24], [31, 16, 37, 24], [31, 18, 35, 24], [31, 20, 33, 24], [31, 22, 31, 24], [31, 24, 29, 24], [31, 26, 27, 24], [31, 28, 25, 24], [33, 0, 53, 22], [33, 2, 51, 22], [33, 4, 49, 22], [33, 6, 47, 22], [33, 8, 45, 22], [33, 10, 43, 22], [33, 12, 41, 22], [33, 14, 39, 22], [33, 16, 37, 22], [33, 18, 35, 22], [33, 20, 33, 22], [33, 22, 31, 22], [33, 24, 29, 22], [33, 26, 27, 22], [33, 28, 25, 22], [33, 30, 23, 22], [35, 0, 53, 20], [35, 2, 51, 20], [35, 4, 49, 20], [35, 6, 47, 20], [35, 8, 45, 20], [35, 10, 43, 20], [35, 12, 41, 20], [35, 14, 39, 20], [35, 16, 37, 20], [35, 18, 35, 20], [35, 20, 33, 20], [35, 22, 31, 20], [35, 24, 29, 20], [35, 26, 27, 20], [35, 28, 25, 20], [35, 30, 23, 20], [35, 32, 21, 20], [37, 0, 53, 18], [37, 2, 51, 18], [37, 4, 49, 18], [37, 6, 47, 18], [37, 8, 45, 18], [37, 10, 43, 18], [37, 12, 41, 18], [37, 14, 39, 18], [37, 16, 37, 18], [37, 18, 35, 18], [37, 20, 33, 18], [37, 22, 31, 18], [37, 24, 29, 18], [37, 26, 27, 18], [37, 28, 25, 18], [37, 30, 23, 18], [37, 32, 21, 18], [37, 34, 19, 18], [39, 0, 53, 16], [39, 2, 51, 16], [39, 4, 49, 16], [39, 6, 47, 16], [39, 8, 45, 16], [39, 10, 43, 16], [39, 12, 41, 16], [39, 14, 39, 16], [39, 16, 37, 16], [39, 18, 35, 16], [39, 20, 33, 16], [39, 22, 31, 16], [39, 24, 29, 16], [39, 26, 27, 16], [39, 28, 25, 16], [39, 30, 23, 16], [39, 32, 21, 16], [39, 34, 19, 16], [39, 36, 17, 16], [41, 0, 53, 14], [41, 2, 51, 14], [41, 4, 49, 14], [41, 6, 47, 14], [41, 8, 45, 14], [41, 10, 43, 14], [41, 12, 41, 14], [41, 14, 39, 14], [41, 16, 37, 14], [41, 18, 35, 14], [41, 20, 33, 14], [41, 22, 31, 14], [41, 24, 29, 14], [41, 26, 27, 14], [41, 28, 25, 14], [41, 30, 23, 14], [41, 32, 21, 14], [41, 34, 19, 14], [41, 36, 17, 14], [41, 38, 15, 14], [43, 0, 53, 12], [43, 2, 51, 12], [43, 4, 49, 12], [43, 6, 47, 12], [43, 8, 45, 12], [43, 10, 43, 12], [43, 12, 41, 12], [43, 14, 39, 12], [43, 16, 37, 12], [43, 18, 35, 12], [43, 20, 33, 12], [43, 22, 31, 12], [43, 24, 29, 12], [43, 26, 27, 12], [43, 28, 25, 12], [43, 30, 23, 12], [43, 32, 21, 12], [43, 34, 19, 12], [43, 36, 17, 12], [43, 38, 15, 12], [43, 40, 13, 12], [45, 0, 53, 10], [45, 2, 51, 10], [45, 4, 49, 10], [45, 6, 47, 10], [45, 8, 45, 10], [45, 10, 43, 10], [45, 12, 41, 10], [45, 14, 39, 10], [45, 16, 37, 10], [45, 18, 35, 10], [45, 20, 33, 10], [45, 22, 31, 10], [45, 24, 29, 10], [45, 26, 27, 10], [45, 28, 25, 10], [45, 30, 23, 10], [45, 32, 21, 10], [45, 34, 19, 10], [45, 36, 17, 10], [45, 38, 15, 10], [45, 40, 13, 10], [45, 42, 11, 10], [47, 0, 53, 8], [47, 2, 51, 8], [47, 4, 49, 8], [47, 6, 47, 8], [47, 8, 45, 8], [47, 10, 43, 8], [47, 12, 41, 8], [47, 14, 39, 8], [47, 16, 37, 8], [47, 18, 35, 8], [47, 20, 33, 8], [47, 22, 31, 8], [47, 24, 29, 8], [47, 26, 27, 8], [47, 28, 25, 8], [47, 30, 23, 8], [47, 32, 21, 8], [47, 34, 19, 8], [47, 36, 17, 8], [47, 38, 15, 8], [47, 40, 13, 8], [47, 42, 11, 8], [47, 44, 9, 8], [49, 0, 53, 6], [49, 2, 51, 6], [49, 4, 49, 6], [49, 6, 47, 6], [49, 8, 45, 6], [49, 10, 43, 6], [49, 12, 41, 6], [49, 14, 39, 6], [49, 16, 37, 6], [49, 18, 35, 6], [49, 20, 33, 6], [49, 22, 31, 6], [49, 24, 29, 6], [49, 26, 27, 6], [49, 28, 25, 6], [49, 30, 23, 6], [49, 32, 21, 6], [49, 34, 19, 6], [49, 36, 17, 6], [49, 38, 15, 6], [49, 40, 13, 6], [49, 42, 11, 6], [49, 44, 9, 6], [49, 46, 7, 6], [51, 0, 53, 4], [51, 2, 51, 4], [51, 4, 49, 4], [51, 6, 47, 4], [51, 8, 45, 4], [51, 10, 43, 4], [51, 12, 41, 4], [51, 14, 39, 4], [51, 16, 37, 4], [51, 18, 35, 4], [51, 20, 33, 4], [51, 22, 31, 4], [51, 24, 29, 4], [51, 26, 27, 4], [51, 28, 25, 4], [51, 30, 23, 4], [51, 32, 21, 4], [51, 34, 19, 4], [51, 36, 17, 4], [51, 38, 15, 4], [51, 40, 13, 4], [51, 42, 11, 4], [51, 44, 9, 4], [51, 46, 7, 4], [51, 48, 5, 4], [53, 0, 53, 2], [53, 2, 51, 2], [53, 4, 49, 2], [53, 6, 47, 2], [53, 8, 45, 2], [53, 10, 43, 2], [53, 12, 41, 2], [53, 14, 39, 2], [53, 16, 37, 2], [53, 18, 35, 2], [53, 20, 33, 2], [53, 22, 31, 2], [53, 24, 29, 2], [53, 26, 27, 2], [53, 28, 25, 2], [53, 30, 23, 2], [53, 32, 21, 2], [53, 34, 19, 2], [53, 36, 17, 2], [53, 38, 15, 2], [53, 40, 13, 2], [53, 42, 11, 2], [53, 44, 9, 2], [53, 46, 7, 2], [53, 48, 5, 2], [53, 50, 3, 2]} ***************** base b = 81 ***************** The base b = 81 has not been pre-computed, but since in this case n = 40: n mod 4 = 0, and so Conjecture A says that N(4,b) is even.Further, Conjecture C says that U = {seq(seq([2*k+1, 2*i, 79-2*i, 80-2*k], i=0..k-1), k=1..39)}, which more explicitly is: {[3, 0, 79, 78], [5, 0, 79, 76], [5, 2, 77, 76], [7, 0, 79, 74], [7, 2, 77, 74], [7, 4, 75, 74], [9, 0, 79, 72], [9, 2, 77, 72], [9, 4, 75, 72], [9, 6, 73, 72], [11, 0, 79, 70], [11, 2, 77, 70], [11, 4, 75, 70], [11, 6, 73, 70], [11, 8, 71, 70], [13, 0, 79, 68], [13, 2, 77, 68], [13, 4, 75, 68], [13, 6, 73, 68], [13, 8, 71, 68], [13, 10, 69, 68], [15, 0, 79, 66], [15, 2, 77, 66], [15, 4, 75, 66], [15, 6, 73, 66], [15, 8, 71, 66], [15, 10, 69, 66], [15, 12, 67, 66], [17, 0, 79, 64], [17, 2, 77, 64], [17, 4, 75, 64], [17, 6, 73, 64], [17, 8, 71, 64], [17, 10, 69, 64], [17, 12, 67, 64], [17, 14, 65, 64], [19, 0, 79, 62], [19, 2, 77, 62], [19, 4, 75, 62], [19, 6, 73, 62], [19, 8, 71, 62], [19, 10, 69, 62], [19, 12, 67, 62], [19, 14, 65, 62], [19, 16, 63, 62], [21, 0, 79, 60], [21, 2, 77, 60], [21, 4, 75, 60], [21, 6, 73, 60], [21, 8, 71, 60], [21, 10, 69, 60], [21, 12, 67, 60], [21, 14, 65, 60], [21, 16, 63, 60], [21, 18, 61, 60], [23, 0, 79, 58], [23, 2, 77, 58], [23, 4, 75, 58], [23, 6, 73, 58], [23, 8, 71, 58], [23, 10, 69, 58], [23, 12, 67, 58], [23, 14, 65, 58], [23, 16, 63, 58], [23, 18, 61, 58], [23, 20, 59, 58], [25, 0, 79, 56], [25, 2, 77, 56], [25, 4, 75, 56], [25, 6, 73, 56], [25, 8, 71, 56], [25, 10, 69, 56], [25, 12, 67, 56], [25, 14, 65, 56], [25, 16, 63, 56], [25, 18, 61, 56], [25, 20, 59, 56], [25, 22, 57, 56], [27, 0, 79, 54], [27, 2, 77, 54], [27, 4, 75, 54], [27, 6, 73, 54], [27, 8, 71, 54], [27, 10, 69, 54], [27, 12, 67, 54], [27, 14, 65, 54], [27, 16, 63, 54], [27, 18, 61, 54], [27, 20, 59, 54], [27, 22, 57, 54], [27, 24, 55, 54], [29, 0, 79, 52], [29, 2, 77, 52], [29, 4, 75, 52], [29, 6, 73, 52], [29, 8, 71, 52], [29, 10, 69, 52], [29, 12, 67, 52], [29, 14, 65, 52], [29, 16, 63, 52], [29, 18, 61, 52], [29, 20, 59, 52], [29, 22, 57, 52], [29, 24, 55, 52], [29, 26, 53, 52], [31, 0, 79, 50], [31, 2, 77, 50], [31, 4, 75, 50], [31, 6, 73, 50], [31, 8, 71, 50], [31, 10, 69, 50], [31, 12, 67, 50], [31, 14, 65, 50], [31, 16, 63, 50], [31, 18, 61, 50], [31, 20, 59, 50], [31, 22, 57, 50], [31, 24, 55, 50], [31, 26, 53, 50], [31, 28, 51, 50], [33, 0, 79, 48], [33, 2, 77, 48], [33, 4, 75, 48], [33, 6, 73, 48], [33, 8, 71, 48], [33, 10, 69, 48], [33, 12, 67, 48], [33, 14, 65, 48], [33, 16, 63, 48], [33, 18, 61, 48], [33, 20, 59, 48], [33, 22, 57, 48], [33, 24, 55, 48], [33, 26, 53, 48], [33, 28, 51, 48], [33, 30, 49, 48], [35, 0, 79, 46], [35, 2, 77, 46], [35, 4, 75, 46], [35, 6, 73, 46], [35, 8, 71, 46], [35, 10, 69, 46], [35, 12, 67, 46], [35, 14, 65, 46], [35, 16, 63, 46], [35, 18, 61, 46], [35, 20, 59, 46], [35, 22, 57, 46], [35, 24, 55, 46], [35, 26, 53, 46], [35, 28, 51, 46], [35, 30, 49, 46], [35, 32, 47, 46], [37, 0, 79, 44], [37, 2, 77, 44], [37, 4, 75, 44], [37, 6, 73, 44], [37, 8, 71, 44], [37, 10, 69, 44], [37, 12, 67, 44], [37, 14, 65, 44], [37, 16, 63, 44], [37, 18, 61, 44], [37, 20, 59, 44], [37, 22, 57, 44], [37, 24, 55, 44], [37, 26, 53, 44], [37, 28, 51, 44], [37, 30, 49, 44], [37, 32, 47, 44], [37, 34, 45, 44], [39, 0, 79, 42], [39, 2, 77, 42], [39, 4, 75, 42], [39, 6, 73, 42], [39, 8, 71, 42], [39, 10, 69, 42], [39, 12, 67, 42], [39, 14, 65, 42], [39, 16, 63, 42], [39, 18, 61, 42], [39, 20, 59, 42], [39, 22, 57, 42], [39, 24, 55, 42], [39, 26, 53, 42], [39, 28, 51, 42], [39, 30, 49, 42], [39, 32, 47, 42], [39, 34, 45, 42], [39, 36, 43, 42], [41, 0, 79, 40], [41, 2, 77, 40], [41, 4, 75, 40], [41, 6, 73, 40], [41, 8, 71, 40], [41, 10, 69, 40], [41, 12, 67, 40], [41, 14, 65, 40], [41, 16, 63, 40], [41, 18, 61, 40], [41, 20, 59, 40], [41, 22, 57, 40], [41, 24, 55, 40], [41, 26, 53, 40], [41, 28, 51, 40], [41, 30, 49, 40], [41, 32, 47, 40], [41, 34, 45, 40], [41, 36, 43, 40], [41, 38, 41, 40], [43, 0, 79, 38], [43, 2, 77, 38], [43, 4, 75, 38], [43, 6, 73, 38], [43, 8, 71, 38], [43, 10, 69, 38], [43, 12, 67, 38], [43, 14, 65, 38], [43, 16, 63, 38], [43, 18, 61, 38], [43, 20, 59, 38], [43, 22, 57, 38], [43, 24, 55, 38], [43, 26, 53, 38], [43, 28, 51, 38], [43, 30, 49, 38], [43, 32, 47, 38], [43, 34, 45, 38], [43, 36, 43, 38], [43, 38, 41, 38], [43, 40, 39, 38], [45, 0, 79, 36], [45, 2, 77, 36], [45, 4, 75, 36], [45, 6, 73, 36], [45, 8, 71, 36], [45, 10, 69, 36], [45, 12, 67, 36], [45, 14, 65, 36], [45, 16, 63, 36], [45, 18, 61, 36], [45, 20, 59, 36], [45, 22, 57, 36], [45, 24, 55, 36], [45, 26, 53, 36], [45, 28, 51, 36], [45, 30, 49, 36], [45, 32, 47, 36], [45, 34, 45, 36], [45, 36, 43, 36], [45, 38, 41, 36], [45, 40, 39, 36], [45, 42, 37, 36], [47, 0, 79, 34], [47, 2, 77, 34], [47, 4, 75, 34], [47, 6, 73, 34], [47, 8, 71, 34], [47, 10, 69, 34], [47, 12, 67, 34], [47, 14, 65, 34], [47, 16, 63, 34], [47, 18, 61, 34], [47, 20, 59, 34], [47, 22, 57, 34], [47, 24, 55, 34], [47, 26, 53, 34], [47, 28, 51, 34], [47, 30, 49, 34], [47, 32, 47, 34], [47, 34, 45, 34], [47, 36, 43, 34], [47, 38, 41, 34], [47, 40, 39, 34], [47, 42, 37, 34], [47, 44, 35, 34], [49, 0, 79, 32], [49, 2, 77, 32], [49, 4, 75, 32], [49, 6, 73, 32], [49, 8, 71, 32], [49, 10, 69, 32], [49, 12, 67, 32], [49, 14, 65, 32], [49, 16, 63, 32], [49, 18, 61, 32], [49, 20, 59, 32], [49, 22, 57, 32], [49, 24, 55, 32], [49, 26, 53, 32], [49, 28, 51, 32], [49, 30, 49, 32], [49, 32, 47, 32], [49, 34, 45, 32], [49, 36, 43, 32], [49, 38, 41, 32], [49, 40, 39, 32], [49, 42, 37, 32], [49, 44, 35, 32], [49, 46, 33, 32], [51, 0, 79, 30], [51, 2, 77, 30], [51, 4, 75, 30], [51, 6, 73, 30], [51, 8, 71, 30], [51, 10, 69, 30], [51, 12, 67, 30], [51, 14, 65, 30], [51, 16, 63, 30], [51, 18, 61, 30], [51, 20, 59, 30], [51, 22, 57, 30], [51, 24, 55, 30], [51, 26, 53, 30], [51, 28, 51, 30], [51, 30, 49, 30], [51, 32, 47, 30], [51, 34, 45, 30], [51, 36, 43, 30], [51, 38, 41, 30], [51, 40, 39, 30], [51, 42, 37, 30], [51, 44, 35, 30], [51, 46, 33, 30], [51, 48, 31, 30], [53, 0, 79, 28], [53, 2, 77, 28], [53, 4, 75, 28], [53, 6, 73, 28], [53, 8, 71, 28], [53, 10, 69, 28], [53, 12, 67, 28], [53, 14, 65, 28], [53, 16, 63, 28], [53, 18, 61, 28], [53, 20, 59, 28], [53, 22, 57, 28], [53, 24, 55, 28], [53, 26, 53, 28], [53, 28, 51, 28], [53, 30, 49, 28], [53, 32, 47, 28], [53, 34, 45, 28], [53, 36, 43, 28], [53, 38, 41, 28], [53, 40, 39, 28], [53, 42, 37, 28], [53, 44, 35, 28], [53, 46, 33, 28], [53, 48, 31, 28], [53, 50, 29, 28], [55, 0, 79, 26], [55, 2, 77, 26], [55, 4, 75, 26], [55, 6, 73, 26], [55, 8, 71, 26], [55, 10, 69, 26], [55, 12, 67, 26], [55, 14, 65, 26], [55, 16, 63, 26], [55, 18, 61, 26], [55, 20, 59, 26], [55, 22, 57, 26], [55, 24, 55, 26], [55, 26, 53, 26], [55, 28, 51, 26], [55, 30, 49, 26], [55, 32, 47, 26], [55, 34, 45, 26], [55, 36, 43, 26], [55, 38, 41, 26], [55, 40, 39, 26], [55, 42, 37, 26], [55, 44, 35, 26], [55, 46, 33, 26], [55, 48, 31, 26], [55, 50, 29, 26], [55, 52, 27, 26], [57, 0, 79, 24], [57, 2, 77, 24], [57, 4, 75, 24], [57, 6, 73, 24], [57, 8, 71, 24], [57, 10, 69, 24], [57, 12, 67, 24], [57, 14, 65, 24], [57, 16, 63, 24], [57, 18, 61, 24], [57, 20, 59, 24], [57, 22, 57, 24], [57, 24, 55, 24], [57, 26, 53, 24], [57, 28, 51, 24], [57, 30, 49, 24], [57, 32, 47, 24], [57, 34, 45, 24], [57, 36, 43, 24], [57, 38, 41, 24], [57, 40, 39, 24], [57, 42, 37, 24], [57, 44, 35, 24], [57, 46, 33, 24], [57, 48, 31, 24], [57, 50, 29, 24], [57, 52, 27, 24], [57, 54, 25, 24], [59, 0, 79, 22], 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48, 31, 6], [75, 50, 29, 6], [75, 52, 27, 6], [75, 54, 25, 6], [75, 56, 23, 6], [75, 58, 21, 6], [75, 60, 19, 6], [75, 62, 17, 6], [75, 64, 15, 6], [75, 66, 13, 6], [75, 68, 11, 6], [75, 70, 9, 6], [75, 72, 7, 6], [77, 0, 79, 4], [77, 2, 77, 4], [77, 4, 75, 4], [77, 6, 73, 4], [77, 8, 71, 4], [77, 10, 69, 4], [77, 12, 67, 4], [77, 14, 65, 4], [77, 16, 63, 4], [77, 18, 61, 4], [77, 20, 59, 4], [77, 22, 57, 4], [77, 24, 55, 4], [77, 26, 53, 4], [77, 28, 51, 4], [77, 30, 49, 4], [77, 32, 47, 4], [77, 34, 45, 4], [77, 36, 43, 4], [77, 38, 41, 4], [77, 40, 39, 4], [77, 42, 37, 4], [77, 44, 35, 4], [77, 46, 33, 4], [77, 48, 31, 4], [77, 50, 29, 4], [77, 52, 27, 4], [77, 54, 25, 4], [77, 56, 23, 4], [77, 58, 21, 4], [77, 60, 19, 4], [77, 62, 17, 4], [77, 64, 15, 4], [77, 66, 13, 4], [77, 68, 11, 4], [77, 70, 9, 4], [77, 72, 7, 4], [77, 74, 5, 4], [79, 0, 79, 2], [79, 2, 77, 2], [79, 4, 75, 2], [79, 6, 73, 2], [79, 8, 71, 2], [79, 10, 69, 2], [79, 12, 67, 2], [79, 14, 65, 2], [79, 16, 63, 2], [79, 18, 61, 2], [79, 20, 59, 2], [79, 22, 57, 2], [79, 24, 55, 2], [79, 26, 53, 2], [79, 28, 51, 2], [79, 30, 49, 2], [79, 32, 47, 2], [79, 34, 45, 2], [79, 36, 43, 2], [79, 38, 41, 2], [79, 40, 39, 2], [79, 42, 37, 2], [79, 44, 35, 2], [79, 46, 33, 2], [79, 48, 31, 2], [79, 50, 29, 2], [79, 52, 27, 2], [79, 54, 25, 2], [79, 56, 23, 2], [79, 58, 21, 2], [79, 60, 19, 2], [79, 62, 17, 2], [79, 64, 15, 2], [79, 66, 13, 2], [79, 68, 11, 2], [79, 70, 9, 2], [79, 72, 7, 2], [79, 74, 5, 2], [79, 76, 3, 2]} ************************************************* ************************************************* 2-Power Conjectures ************************************************* ************************************************** For a b >= 4 which is a power of 2, let n be the unique integer s.t. b = 2^n.Then we have the following conjectures: n-even conjecture: N(4,b) = n. Specifically, writing n = 2k we have that L(4,b) = [k, k+1, seq(2k, i=1..k-1), seq(2(k+1), i=1..k-1)]. n-odd conjecture: N(4,b) = n-1. Specifically, writing n = 2k+1 we have that L(4,b) = [seq(2k+1, i=1..k), seq(2(k+1)+1, i=1..k)] ***************** base b = 4 ***************** For b = 4, so n = 2, and in particular n is even, and k = 1. The list of all limiting orbits is: [[3, 0, 2, 1]] [[1, 3, 3, 2], [2, 0, 2, 2]] Thus N(4,b) = 2 and L(4,b) = [1, 2]. Thus the n-even Conjecture is true. ***************** base b = 16 ***************** For b = 16, so n = 4, and in particular n is even, and k = 2. The list of all limiting orbits is: [[5, 2, 12, 11], [10, 5, 9, 6]] [[3, 15, 15, 12], [12, 2, 12, 4], [10, 7, 7, 6]] [[3, 0, 14, 13], [14, 9, 5, 2], [12, 3, 11, 4], [9, 6, 8, 7]] [[1, 15, 15, 14], [14, 0, 14, 2], [14, 11, 3, 2], [12, 7, 7, 4], [7, 15, 15, 8], [8, 6, 8, 8]] Thus N(4,b) = 4 and L(4,b) = [2, 3, 4, 6]. Thus the n-even Conjecture is true. ************************************************* ************************************************* 3-Power Conjectures ************************************************* ************************************************** For a base b >= 9 which is a power of 3, let n be the unique integer s.t. b = 3^n. Then we have the following conjecture: Conjecture: N(4,b) = sum(3^i-1, i=1..n-1). In particular L(4,b) = [seq(seq(3^k, i=1..3^k-1), k=1..n-1)]. ***************** base b = 81 ***************** The base b = 81 has not been pre-computed, but since in this case n = 4: the conjecture says that N(4,b) = 36 and L(4,b) = [3, 3, 9, 9, 9, 9, 9, 9, 9, 9, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27]. ************************************************* ************************************************* 3x2-Power Conjectures ************************************************* ************************************************** For a b >= 6 which is of the form b = 3*2^n, we have the following conjectures: n-even conjecture: N(4,b) = 2. Specifically, writing n = 2k we have that L(4,b) = [2k+1, 2(2k+1)].In particular, the first orbit is generated by [2^n - 1, 3*2^n - 1, 3*2^n - 1, 2^(n+1)]and the second orbit is generated by [2^n, 2^(n-1)-1, 5*2^(n-1) - 1, 2^(n+1)]. n-odd conjecture: N(4,b) = 1. Specifically, writing n = 2k+1 we have that L(4,b) = [6(k+1)].In particular, the singular orbit is generated by [2^n, 2^(n-1)-1, 5*2^(n-1) - 1, 2^(n+1)]. ***************** base b = 24 ***************** For b = 24, n = 3, and in particular n is odd, and k = 1. The list of all limiting orbits is: [[7, 23, 23, 16], [16, 6, 16, 8], [10, 7, 15, 14], [8, 3, 19, 16], [16, 7, 15, 8], [9, 6, 16, 15], [10, 5, 17, 14], [12, 3, 19, 12], [15, 23, 23, 8], [15, 7, 15, 9], [8, 5, 17, 16], [12, 7, 15, 12]] Thus N(4,b) = 1 and L(4,b) = [12]. Further, the generator for the limiting orbit is [7, 23, 23, 16]. Thus the n-odd Conjecture is true. ******************************************* * * Length 7 conjecture(s) for bases * {4, 16, 18, 24, 55, 81} * ******************************************* For a given base b and number of digits l, let N(l,b) denote the number of limiting orbits occuring over all possible input numbers num with l digits in base b to the iterative procedure listPhenom(num,b), and let L(l,b) denote the list of lengths of each orbit, sorted in increasing order. ************************************************* ************************************************* Even-Base Conjecture(s) ************************************************* ************************************************* For an even b >= 2, we have the following conjectures: b mod 4 = 0 conjecture: Writing b = 4n, if n >= 6 we have that: N(7,b) = 2 and L(7,b) = [1, 19]. In particular, length 1 orbit is given by [[3*n, 2*n, n-1, 4*n-1, 3*n-1, 2*n-1, n]], and the length 19 orbit is generated by [3*n-1, 2*n-2, n-2, 4*n-1, 3*n, 2*n+1, n+1]. b mod 4 = 2 conjecture: Writing b = 4n+2, if n >= 4 we have that: N(7,b) = 1 and L(7,b) = [4]. In particular, the singular orbit is generated by [3*n+1, 2*n, n-2, 4*n+1, 3*n+2, 2*n+1, n+1]. ***************** base b = 18 ***************** For b = 18, n = 4, and in particular b mod 4 = 2. The list of all limiting orbits is: [[13, 8, 2, 17, 14, 9, 5], [15, 9, 4, 17, 12, 8, 3], [14, 11, 3, 17, 13, 6, 4], [14, 10, 6, 17, 10, 7, 4]] Thus N(7,b) = 1 and L(7,b) = [4]. Further, the generator for the limiting orbit is [13, 8, 2, 17, 14, 9, 5]. Thus the b mod 4 = 2 conjecture is true. ***************** base b = 24 ***************** The base b = 24 has not been pre-computed, but since in this case n = 6, and in particular b mod 4 = 0: the b mod 4 = 0 conjecture says that N(7,b) = 2 and L(7,b) = [1, 19].Further, it says that the length 1 orbit is [[18, 12, 5, 23, 17, 11, 6]] and the generator for the length 19 orbit is [17, 10, 4, 23, 18, 13, 7]. This took 0.065000 seconds.