Pollock octahedral numbers conjecture

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The Pollock octahedral numbers conjecture is a conjecture that every integer is the sum of at most seven octahedral numbers, first stated by the mathematician F. Pollock. (Dickson 2005, p. 23).

Sergio Demian Lerner (2005) has conjectured that there is an upper bound on numbers requiring 5 terms or more. He also conjectures this upper bound is 65285683, which requires 5 terms. The findings are based on the brute force enumeration and decomposition of the first 300 million integers. Computer results show that:

309 seems to be the last number requiring 7 terms
11579 seems to be the last number requiring 6 terms
65285683 seems to be the last number requiring 5 terms.

If the conjecture is true, then it can be easily proven that there must be arbitrary high numbers requiring 4 terms.

References

Dickson, L. E., History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005.
Sergio Demian Lerner, unpublished paper.
Eric W. Weisstein. "Octahedral Number." From MathWorld--A Wolfram Web Resource.[link]

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