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Largest number of primes

Source: Baltic Way 2021, Problem 17

November 15, 2021
number theorynumber theory proposedprimes

Problem Statement

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?
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