Pollock's Conjecture

Author

Eric W. Weisstein
February 26, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/PollocksConjecture.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PollocksConjecture.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Greedy Algorithm

TetrahedralNumber/@Range[10]

Roots[TetrahedralNumber[n]==k,n,Cubics->False]

Table[Min[Plus@@@GreedyAll[TetrahedralNumber/@Range[Root[-6 n+2 #1+3 ((#1)^2)+((#1)^3)&,1]],n]],{n,20}]

Position[l,_?(#>4&)]//Flatten

Not Expressible by ≤4 Tetrahedral Numbers

l={17,27,33,52,73,82,83,103,107,137,153,162,217,219,227,237,247,258,268,271,282,283,302,303,313,358,383,432,437,443,447,502,548,557,558,647,662,667,709,713,718,722,842,863,898,953,1007,1117,1118,1153,1227,1233,1243,1314,1382,1402,1468,1478,1513,1523,1578,1612,1622,1658,1678,1693,1731,1738,1742,1758,1767,1803,1858,1907,1923,1933,2037,2053,2172,2198,2217,2218,2251,2253,2327,2372,2382,2417,2437,2457,2537,2538,2578,2687,2818,2858,2898,2973,3138,3142,3183,3218,3263,3463,3512,3887,4003,4307,4317,4563,4832,4923,5013,5142,5238,5283,5483,5508,5538,5563,5618,5647,5707,6022,6057,6067,6186,6213,6263,6343,6462,6863,7067,7278,7377,7387,7423,7497,7542,7662,7793,7873,8223,8307,8322,8973,9063,9488,9687,9753,9772,9973,10397,10467,10532,10633,10852,11237,11302,11737,11962,12247,12547,12722,12777,12843,12858,13127,13393,13822,14492,15122,15483,15867,16097,16538,16637,16742,17253,17683,17813,17893,18573,18782,19168,19277,20918,21523,22618,22657,23677,24237,24317,24338,25447,25723,26007,27858,28617,28847,29157,29487,29938,30298,31973,33183,36262,36913,37798,38453,38707,38807,39693,39913,41278,41322,41433,44833,47627,48043,56467,56842,58613,59077,62158,64752,65253,65567,71157,74687,78003,78787,83603,84023,85993,91128,106277,113062,134038,148437,343867};