Abstract: We show that there are infinitely many primes of the form X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2.
Our work builds on the famous Friedlander-Iwaniec result on primes of the form X^2+Y^4.
More precisely, Friedlander and Iwaniec obtained an asymptotic formula for the number of primes of this form.
For the argument we need to estimate Type II sums, which is achieved by an application of the Weil bound,
both for point-counting and for exponential sums over curves. The type II information we get is too narrow
for an asymptotic formula, but we can apply Harman's sieve method to establish a lower bound of the
correct order of magnitude for the number of primes of the form X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2.
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