For s ≥ 3k+2 the prime Hasse principle holds for almost all equations a1 x1^k + ... + as xs^k = 0.
Hu, The probability that random positive integers are k-wise relatively prime , Inter- national Journal of Number Theory 9 (2013), 1263-1271
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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.