# 285 CE

## Diophantine equation

## The book of science

Tom Sharp

Tom Sharp

Diophantus mathematics |

- 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.

- 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 a

^{2}+ b^{2}= c^{2}where a < b.The first taxicab number, 1

^{3}+ 12^{3}= 9^{3}+ 10^{3}= 1729.Pell numbers are integer solutions to x

^{2}- 2y^{2}= ±1.See also in

The book of science:Proof of the Pythagorean theorem—PythagorasReadings in wikipedia: