For s ≥ 3k+2 the prime Hasse principle holds for almost all equations a1 x1^k + ... + as xs^k = 0.
T´ oth, The probability that k positive integers are pairwise relatively coprime Fi- bonacci Quarterly 40 (2002), 13-18
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Random Diophantine Equations in the Primes
For s ≥ 3k+2 the prime Hasse principle holds for almost all equations a1 x1^k + ... + as xs^k = 0.