VTMO 2026 - Grade 6 - Final - SET 5 - Self-study Question 12

Find all pairs of positive integers $a,b$ satisfying both conditions: $a,a+2b,a+4b,a+8b,a+16b$ are prime numbers; and $b,b+2a,b+4a,b+8a,b+16a$ are prime numbers. Give a complete proof.