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title. I Tried using Gaussian integers but it got me nowhere. I'm not even sure a 'nice' characterization is possible.

Mkch
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Basically, we are looking for primes $p=n^2 + (n+1)^2$ according to Wojowu. But primes, except the first two, are either of the form $p=6k+1$ or $p=6k-1$. So we are looking for solutions of $2n^2 + 2n + 1 = 6k-1$ and $2n^2 +2n +1 = 6k+1$. So the problem is now reduced to solving a quadratic equation in $n$ with a parameter $k$. We demand that the discriminant $d$ of these quadratic equations be a square.

The case of $p=6k-1$ has a $d=12k-3$. $d$ is a square for $k=1,7,19,...$ and the corresponding primes are $p=5=1^2 + 2^2$, $p=41=4^2 + 5^2$ and $p=113=7^2 +8^2$. Most probably, not all square values of d will lead to a prime.

The case of $p=6k+1$ has a $d=12k+1$. $d$ is a square for $k=2,10,30,52...$ and the primes are $p=13= 2^2 + 3^2$, $p=61=5^2 + 6^2$, $p=181=9^2 + 10^2$, $p=313=12^2 +13^2$.

It would be nice if someone could write a program (I cannot) to look for more $k$ values that lead to primes. By the way, the first two primes cannot be written as a sum of two consecutive primes.

user25406
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    There are lots and lots. The first few are 5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681, 30013, 34061, 36721, 37813, 38921, 41761, 44701, 47741, 49613, 51521, 52813, 54121, 56113, 59513, 60901, 68821, 70313, 71821, 76441, 79601, 82013, 83641, 86113, 90313, 94613, 97241, 99013, 100801, 101701, 105341, 106261, 110921, 134681, 135721, 139921, 143113, 148513, 156241, 161881, 163021, 165313, 167621, … – Patrick Stevens Jan 30 '18 at 20:03
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    I think the op was looking for a method he could use to generate these primes. – user25406 Jan 31 '18 at 19:18