A100878 Smallest number of pentagonal numbers which sum to n.

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

Offset: 0

Author

Franz Vrabec, Jan 09 2005

Keywords

• nonn

Comments

From Bernard Schott, Jul 15 2022: (Start)

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)

Examples

			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.
		

References

• 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.

Links

• Augustin-Louis Cauchy, Démonstration du théorème général de Fermat sur les nombres polygones, Extrait des Mémoires de l'Institut, 1813-15.
• Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem.
• Wikipedia, Fermat polygonal number theorem.

Crossrefs

Cf. A002828, A061336, A355717.

Cf. A000326 (a(n) = 1), A003679 (a(n) = 4 or 5), A355660 (a(n) = 4), A133929 (a(n) = 5).

Programs

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

Formula

a(n) <= 5 (inequality proposed by Fermat and proved by Cauchy). - Bernard Schott, Jul 13 2022

Extensions

More terms from David Wasserman, Mar 04 2008