Indeterminate linear problems from Asia to Europe

Abstract

In the Chinese tradition, linear Diophantine problems can be traced back to two main categories. The first group includes the problems consisting of 2-equation systems with integer coefficients in n unknowns with $$n> 2$$ and integer solutions, such as: $$x_1 + x_2 + \dots + x_n = p$$ , $$b_1 x_1 + b_2 x_2+ \dots + b_n x_n = q$$ . The so-called “100 fowls problem” belongs to this group. The oldest statement can apparently be found in the Zhang Qiujian Suanjing (The Computation Classic of Zhang Qiujian), towards the second half of the 5th century AD (468–486). In the second category there are problems that can be represented through simultaneous linear congruences, which are generally formulated as follows: find a number x that, divided by $$m_1$$ , $$m_2$$ , $$m_3, \ldots , m_i$$ , give as remainders $$r_1$$ , $$r_2$$ , $$r_3, \ldots , r_i$$ . This kind of problem is currently called “Chinese (remainder) problem”. The oldest formulation is attributed to Master Sun Tzu (between 280 and 473). We shall present the different statements of the two problems in their circulation from Asia to Europe and the main proof procedures.