History
The mathematical study of figurate numbers is said to have originated with Pythagoras, possibly based on Babylonian or Egyptian precursors. Generating whichever class of figurate numbers the Pythagoreans studied using gnomons is also attributed to Pythagoras. Unfortunately, there is no trustworthy source for these claims, because all surviving writings about the Pythagoreans are from centuries later. It seems to be certain that the fourth triangular number of ten objects, called tetractys in Greek, was a central part of the Pythagorean religion, along with several other figures also called tetractys. Figurate numbers were a concern of Pythagorean geometry.
The modern study of figurate numbers goes back to Fermat, specifically the Fermat polygonal number theorem. Later, it became a significant topic for Euler, who gave an explicit formula for all triangular numbers that are also perfect squares, among many other discoveries relating to figurate numbers.
Figurate numbers have played a significant role in modern recreational mathematics. In research mathematics, figurate numbers are studied by way of the Ehrhart polynomials, polynomials that count the number of integer points in a polygon or polyhedron when it is expanded by a given factor.
Read more about this topic: Figurate Number
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