Illustration of Diophantine equation

285 CE Diophantine equation

The book of science

Tom Sharp

Diophantus mathematics Illustration of Diophantine equation

Diophantine equation

A diophantine equation has more than one unknown. We look for only integer solutions. We are interested in integers, whole numbers, not fractions or continuous lines. * Diophantine solutions of the Pythagorean theorem are Pythagorean triples, (3,4,5), (5,12,13), and so forth. Taxicab numbers, starting with 1729, can be expressed as the sum of the same number of cubes in two different ways. Whatever you multiply one square so the difference from another square is 1 are Pell numbers. These are the denominators of the closest rational approximations to the square root of 2.

Pell hadn’t solved it

Brahmagupta showed how to solve it in 628. Bhāskara II solved it in 1150. Narayana Pandit solved it in 1356. Pierre de Fermat and William Brouncker solved it in 1657 but Leonhard Euler named it after John Pell.


The number of unknowns is greater than the number of equations. The number of ways to answer each question is greater than the number of questions. Wrong ways are more numerous, but for many things there are no wrong ways. A series of correct answers never reaches an imagined convergence.

Pythagorean triples are integer solutions to a2 + b2 = c2 where a < b.

The first taxicab number, 13 + 123 = 93 + 103 = 1729.

Pell numbers are integer solutions to x2 - 2y2 = ±1.

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