MathDB

17

Part of 2021 Baltic Way

Problems(1)

Largest number of primes

Source: Baltic Way 2021, Problem 17

11/15/2021
Distinct positive integers a,b,c,da, b, c, d satisfy {a∣b2+c2+d2,b∣a2+c2+d2,c∣a2+b2+d2,d∣a2+b2+c2,\begin{cases} a \mid b^2 + c^2 + d^2,\\ b\mid a^2 + c^2 + d^2,\\ c \mid a^2 + b^2 + d^2,\\ d \mid a^2 + b^2 + c^2,\end{cases} and none of them is larger than the product of the three others. What is the largest possible number of primes among them?
number theorynumber theory proposedprimes