For s ≥ 3k+2 the prime Hasse principle holds for almost all equations a1 x1^k + ... + as xs^k = 0.
Koukoulopoulos, The Distribution of Prime Numbers , Graduate Studies in Mathe- matics 203, American Mathematical Society, 2019
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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.