Integer as Sum of Polygonal Numbers/Historical Note

Historical Note on Integer as Sum of Polygonal Numbers

Many of Fermat's theorems were stated, mostly without proof, in the margin of his copy of Bachet's translation of Diophantus's Arithmetica.

In $1670$, his son Samuel published an edition of this, complete with Fermat's marginal notes.

Fermat's Note

As Fermat himself put it:

Every positive integer is triangular or the sum of $2$ or $3$ triangular numbers; a square or the sum of $2$, $3$ or $4$ squares; a pentagonal number or the sum of $2$, $3$, $4$ or $5$ pentagonal numbers; and so on to infinity, whether it is a question of hexagonal, heptagonal or any polygonal numbers.

I cannot give the proof here, for it depends on many abstruse mysteries of numbers; but I intend to devote an entire book to this subject, and to present in this part of number theory astonishing advances beyond previously known boundaries.

Euler struggled on and off with the proof for squares without success for nearly $40$ years. Lagrange finally managed this in $1772$.

Gauss proved the theorem for triangular numbers in $1796$.

Finally, Cauchy proved the theorem in its completeness in $1815$.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$)