How to solve the problem that determines the age of Diophantus?$$$$"God gave him to be a boy for the sixth part of his life, and adding one to it twelfth part covered her cheeks fluff, He gave him the bridal lamp after a seventh part, and five years after his marriage gave him a son . Ai! Unhappy child late, after arriving as half of his father's life, chill Fate took him. After consoling his grief over four years with the science of numbers he ended his life." (Boyer, 1996, p. 130)$$$$
2026-04-09 09:11:04.1775725864
How to solve the problem that determines the age of Diophantus?
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Let $D$ be the number of years Diophantus lived, and $S$ the number of years his son lived. First we make an obvious relation, that his son lived for half his own lifetime. $$S = \dfrac{D}{2}$$ Next we use all the other details, $$D = \left[\dfrac{D}{6}+\dfrac{D}{12}+\dfrac{D}{7} + 5\right]+ S + 4$$ that is, the length of his life equals the time until his son was born (bracketed), plus his son's life and an extra four years. You have two equations and two unknowns. Take it from here.