Solutions of the Diophantine equation nx + (7p)y = z2

Key Points

• Integer solutions for nx + (7p)y = z² exist under specific conditions concerning primes.
• The exploration focuses on positive n and prime p, with x, y, and z taking non-negative values.
• Systematic checks evaluate possible integer combinations to confirm valid outcomes.
• This work highlights the need for thorough exploration of integer solution criteria.