## Pollock octahedral numbers conjecture.

For the discussion of math. Duh.

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### Pollock octahedral numbers conjecture.

In additive number theory, the Pollock octahedral numbers conjecture is an unproven conjecture that every positive integer is the sum of at most seven octahedral numbers.

Ripped this from wikipedia. wondering if anyone here would be able to prove or disprove it.

Yes, my name is the negative frequency of radioactive decay, of the initial speed of light's radius.
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.

krucifi

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### Re: Pollock octahedral numbers conjecture.

krucifi wrote:Ripped this from wikipedia. wondering if anyone here would be able to prove or disprove it.
It's been unproven for 162 years. I'm going to go out on a limb here and guess that no one's going to prove or disprove it in a forum thread.
In the future, there will be a global network of billions of adding machines.... One of the primary uses of this network will be to transport moving pictures of lesbian sex by pretending they are made out of numbers.
Spoiler:
gmss1 gmss2

gmalivuk
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### Re: Pollock octahedral numbers conjecture.

gmalivuk wrote:It's been unproven for 162 years. I'm going to go out on a limb here and guess that no one's going to prove or disprove it in a forum thread.

FWIW, here's a simple Python program that attempts to express integers as the sum of at most seven octahedral numbers using a simple greedy algorithm, and prints those integers which it fails on. As a bonus, it has an inverse of the octahedral number function.
Code: Select all
`#! /usr/bin/env python''' Investigating the Pollock octahedral numbers conjecture: every positive integer is the sum of at most seven octahedral numbers. Find numbers that can't be expressed as such a sum by the greedy algorithm. See http://en.wikipedia.org/wiki/Pollock_octahedral_numbers_conjecture '''import sys, mathfrom bisect import bisect#Calculate the nth octahedral numberdef octa(n): return n * (2*n*n + 1) / 3#Invert octa() using u**3 + v**3 = (u+v)**3 - 3*u*v*(u+v)def invocta(y):    d = math.sqrt(729 * y * y + 6)    w = (27 * y + d) / 36.    u = math.pow(w, 1. / 3)    x = u - 1. / (6 * u)        #Perform one round of Newton's method to reduce error    #x2 = x * x     ##x += (3 * y -  x * (2 * x2 + 1)) / (6 * x2 + 1)     #x = (3 * y + 4 * x * x2) / (6 * x2 + 1)    return xdef pollock(m):    octalist = [octa(i) for i in xrange(1, m + 1)]    octamax = octalist[-1]    print >>sys.stderr, 'Searching up to %d = octa(%d)' % (octamax, m)    #Find highest j, octa(j): octa(j) < i, j<=hi     def octaceil(i, hi=m):        j = bisect(octalist, i, hi=hi)                return j, octalist[j-1]    #See if we can produce i as a sum of octahedral numbers <= octa(hi)    def greedy(i, hi):        for k in xrange(7):            j, v = octaceil(i, hi)            hi = j            i -= v            if i == 0:                 return True        return False    #Find octahedral sets that sum to i    count = 0    for i in xrange(1, octamax + 1):        if not greedy(i, m):            count += 1            print '%d,' % i,        if (i + 1) % 50 == 0:            print    print    print >>sys.stderr, '%d exceptions found. Ratio=%f' % (count, float(count) / octamax)def main():    m = len(sys.argv) > 1 and int(sys.argv[1]) or 5    pollock(m)    if __name__ == '__main__':    main()`

Here's some output:
Code: Select all
`Searching up to 2255 = octa(15)36,61, 74, 79, 80,102, 115, 120, 121, 128, 140, 145,163, 176, 181, 182, 189,201, 206, 207, 214, 219, 220, 224, 225, 226, 248,261, 266, 267, 274, 286, 291, 292, 299,304, 305, 309, 310, 311, 327, 332, 333, 340,361, 374, 379, 380, 387, 399,404, 405, 412, 417, 418, 422, 423, 424, 440, 445, 446,453, 458, 459, 463, 464, 465, 471, 472, 478, 483, 484, 488,506, 519, 524, 525, 532, 544, 549,550, 557, 562, 563, 567, 568, 569, 585, 590, 591, 598,603, 604, 608, 609, 610, 616, 617, 623, 628, 629, 633, 634, 646,651, 652, 659, 664, 665, 669, 687,700, 705, 706, 713, 725, 730, 731, 738, 743, 744, 748, 749,750, 766, 771, 772, 779, 784, 785, 789, 790, 791, 797, 798,804, 809, 810, 814, 815, 827, 832, 833, 840, 845, 846,850, 851, 852, 858, 859, 865, 870, 871, 875, 876, 877, 883, 884, 888, 889, 890,908, 921, 926, 927, 934, 946,951, 952, 959, 964, 965, 969, 970, 971, 987, 992, 993,1000, 1005, 1006, 1010, 1011, 1012, 1018, 1019, 1025, 1030, 1031, 1035, 1036, 1048,1053, 1054, 1061, 1066, 1067, 1071, 1072, 1073, 1079, 1080, 1086, 1091, 1092, 1096, 1097, 1098,1104, 1105, 1109, 1110, 1111, 1114, 1115, 1116, 1117, 1133, 1138, 1139, 1146,1151, 1152, 1173, 1186, 1191, 1192, 1199,1211, 1216, 1217, 1224, 1229, 1230, 1234, 1235, 1236,1252, 1257, 1258, 1265, 1270, 1271, 1275, 1276, 1277, 1283, 1284, 1290, 1295, 1296,1300, 1301, 1313, 1318, 1319, 1326, 1331, 1332, 1336, 1337, 1338, 1344, 1345,1351, 1356, 1357, 1361, 1362, 1363, 1369, 1370, 1374, 1375, 1376, 1379, 1380, 1381, 1382, 1398,1403, 1404, 1411, 1416, 1417, 1421, 1422, 1423, 1429, 1430, 1436, 1441, 1442, 1446, 1447, 1448,1454, 1455, 1459, 1460, 1461, 1464, 1465, 1466, 1467, 1486, 1499,1504, 1505, 1512, 1524, 1529, 1530, 1537, 1542, 1543, 1547, 1548, 1549,1565, 1570, 1571, 1578, 1583, 1584, 1588, 1589, 1590, 1596, 1597,1603, 1608, 1609, 1613, 1614, 1626, 1631, 1632, 1639, 1644, 1645, 1649,1650, 1651, 1657, 1658, 1664, 1669, 1670, 1674, 1675, 1676, 1682, 1683, 1687, 1688, 1689, 1692, 1693, 1694, 1695,1711, 1716, 1717, 1724, 1729, 1730, 1734, 1735, 1736, 1742, 1743, 1749,1754, 1755, 1759, 1760, 1761, 1767, 1768, 1772, 1773, 1774, 1777, 1778, 1779, 1780, 1790, 1795, 1796,1800, 1801, 1802, 1808, 1809, 1824, 1829, 1830,1851, 1864, 1869, 1870, 1877, 1889, 1894, 1895,1902, 1907, 1908, 1912, 1913, 1914, 1930, 1935, 1936, 1943, 1948, 1949,1953, 1954, 1955, 1961, 1962, 1968, 1973, 1974, 1978, 1979, 1991, 1996, 1997,2004, 2009, 2010, 2014, 2015, 2016, 2022, 2023, 2029, 2034, 2035, 2039, 2040, 2041, 2047, 2048,2052, 2053, 2054, 2057, 2058, 2059, 2060, 2076, 2081, 2082, 2089, 2094, 2095, 2099,2100, 2101, 2107, 2108, 2114, 2119, 2120, 2124, 2125, 2126, 2132, 2133, 2137, 2138, 2139, 2142, 2143, 2144, 2145,2155, 2160, 2161, 2165, 2166, 2167, 2173, 2174, 2189, 2194, 2195,2202, 2207, 2208, 2212, 2213, 2214, 2220, 2221, 2227, 2232, 2233, 2237, 2238, 2239, 2245, 2246,2250, 2251, 2252,511 exceptions found. Ratio=0.226608`

As we search higher integers the success ratio appears to decrease:
Code: Select all
`Searching up to 670 = octa(10)110 exceptions found. Ratio=0.164179Searching up to 5340 = octa(20)1424 exceptions found. Ratio=0.266667Searching up to 18010 = octa(30)5808 exceptions found. Ratio=0.322488Searching up to 42680 = octa(40)15320 exceptions found. Ratio=0.358950Searching up to 83350 = octa(50)32176 exceptions found. Ratio=0.386035Searching up to 144020 = octa(60)58637 exceptions found. Ratio=0.407145Searching up to 228690 = octa(70)97028 exceptions found. Ratio=0.424277Searching up to 341360 = octa(80)149849 exceptions found. Ratio=0.438976Searching up to 486030 = octa(90)219618 exceptions found. Ratio=0.451861Searching up to 666700 = octa(100)308779 exceptions found. Ratio=0.463145Searching up to 1302125 = octa(125)633237 exceptions found. Ratio=0.486310Searching up to 2250050 = octa(150)1134999 exceptions found. Ratio=0.504433Searching up to 3572975 = octa(175)1855935 exceptions found. Ratio=0.519437Searching up to 5333400 = octa(200)2837496 exceptions found. Ratio=0.532024`

Edit: Arrrgh! I had a stupid sign error under the square root in the invocta() function.
Last edited by PM 2Ring on Thu May 03, 2012 2:09 pm UTC, edited 1 time in total.

PM 2Ring

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### Re: Pollock octahedral numbers conjecture.

gmalivuk wrote:
krucifi wrote:Ripped this from wikipedia. wondering if anyone here would be able to prove or disprove it.
It's been unproven for 162 years. I'm going to go out on a limb here and guess that no one's going to prove or disprove it in a forum thread.
Actually, this problem looks quite approachable. Using the circle method, you can show that every sufficiently large number is a sum of at most seven cubes, and octahedral numbers are more common than the cubes, and appear to have fewer congruence conditions placed on them. It's possible that the real reason it hasn't been solved yet is simply that no one has tried applying the circle method to it recently.
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notzeb
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### Re: Pollock octahedral numbers conjecture.

I haven't yet learned enough about the circle method to tackle the problem theoretically but I have made empirical progress on a related problem.

For a while now I've been working on Pollock's (Tetrahedral) Conjecture and I think I've made some good progress. Previously published distribution statistics go up to 40B integers and now I've pushed that up to 400B. If you want to read more about it I have a blog post with links to the code at http://euler.rouxheyns.com (I apparently can't make links yet )

If you want to apply my technique to this problem to get some numerical insight you're welcome to use the code and I'll be happy to answer any questions you have.
rheyns

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