"Find a quantity such that if its quarter is added to it, it becomes 15": this problem was posed in the Rhind Papyrus 3,600 years ago. The solution is simple: since 4 plus a quarter of itself is 5, multiplying by 3 gives 15—and hence the answer, 12. Just as Molière's Mr Jourdain spoke prose without knowing it, we have all encountered equations of this kind, known as Diophantine equations, without realizing it. They are named after the great Greek arithmetician Diophantus of Alexandria (between 150 and 350 CE): equations with integer coefficients for which integer solutions are sought.

An excerpt from the Rhind Papyrus.

Such equations were also studied by Euclid (in the 3rd century BCE), who laid the foundations of arithmetic, notably with the lemma that bears his name ("If a prime number p divides the product bc of two integers, then p divides b or c"). This result would encourage the search for Pythagorean triples (see our feature in Tangente 212, 2023), integer solutions of the equation *x 2 + *y 2 = *z 2, another Diophantine equation!
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