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A100878
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Smallest number of pentagonal numbers which sum to n.
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10
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0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 3, 4, 2, 1, 2, 3, 4, 3, 2, 3, 4, 5, 2, 3, 3, 2, 3, 3, 4, 1, 2, 3, 4, 5, 2, 2, 3, 3, 4, 3, 3, 2, 3, 4, 3, 4, 3, 3, 1, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 4, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 4, 2, 3, 2
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OFFSET
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0,3
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COMMENTS
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In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers, four squares, five pentagonal numbers, and so on.
The square case was proved by Lagrange in 1770; it is known as Lagrange's four squares theorem (see A002828). Then Gauss proved the triangular case in 1796 (see A061336).
In 1813, Cauchy proved this polygonal number theorem: for m >= 3, every positive integer N can be represented as a sum of m+2 (m+2)-gonal numbers, at most four of which are different from 0 and 1 (Deza reference). Hence every number is expressible as the sum of at most five positive pentagonal numbers (A000326). (End)
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REFERENCES
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Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.
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LINKS
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FORMULA
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a(n) <= 5 (inequality proposed by Fermat and proved by Cauchy). - Bernard Schott, Jul 13 2022
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EXAMPLE
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a(5)=1 since 5=5, a(6)=2 since 6=1+5, a(7)=3 since 7=1+1+5, a(10)=2 since 10=5+5 with 1 and 5 pentagonal numbers.
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PROG
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(PARI) a(n) = my(nb=oo); forpart(vp=n, if (vecsum(apply(x->ispolygonal(x, 5), Vec(vp))) == #vp, nb = min(nb, #vp)), , 5); nb; \\ Michel Marcus, Jul 15 2022
(PARI) a(n) = for(i = 1, oo, p = partitions(n, , [i, i]); for(j = 1, #p, if(sum(k = 1, i, ispolygonal(p[j][k], 5)) == i, return(i)))) \\ David A. Corneth, Jul 15 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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