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Can we represent any prime $p \equiv 1 \pmod{5}$ or $p \equiv -1 \pmod{5}$ in terms of $(a+b)^2 + ab$ with $a> b> 0$?

Can we represent any prime $p \equiv 1 \pmod{3}$ in terms of $(a+b)^2 - ab$ with $a> b> 0$?

can we have, any odd integer $> 1$ can be written as $a + b$ where $a$ and $b$ are positive integers with $a^4$ + $b^4$ is prime?

Kindly discuss. Thanks in advance

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    And what I am looking for from you is an answer to the question, what do you mean by "I am searching by computations in different ways"? The $p\equiv1\pmod3$ thing can be found in many intro number theory texts.2012-11-08

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As another answer shows, if $p\equiv1\pmod3$, then $-3$ is a quadratic residue modulo $p$. That means there is an integer $x$ such that $p{\rm\ divides\ }x^2+3$ Now we pass to ${\cal O}_{-3}$, the ring of integers in the number field $K={\bf Q}(\sqrt{-3})$, and write, $p{\rm\ divides\ }(x+\sqrt{-3})(x-\sqrt{-3})$ Now $p$ doesn't divide either one of $x\pm\sqrt{-3}$ since ${x\over p}\pm{1\over p}\sqrt{-3}$ are not in ${\cal O}_{-3}$. But it is known that ${\cal O}_{-3}$ is a unique factorization domain, implying that if an irreducible divides a product it must divide one of the factors. We deduce that $p$ is not an irreducible in ${\cal O}_{-3}$. Let $\pi=a+b{1+\sqrt{-3}\over2}$ be a nontrivial factor of $p$ in ${\cal O}_{-3}$. Then the norm of $\pi$ is a positive nontrivial factor of the norm of $p$, which is $p^2$, so the norm of $\pi$ is $p$. But the norm of $\pi$ is $a^2+ab+b^2$, QED.

EDIT: Similarly, if $p\equiv\pm1\pmod5$, then $5$ is a quadratic residue modulo $p$, so $p{\rm\ divides\ }x^2-5$ for some $x$. Going to ${\cal O}_5$, $p{\rm\ divides\ }(x+\sqrt5)(x-\sqrt5)$ but $p$ divides neither one of $x\pm\sqrt5$. Now ${\cal O}_5$ is known to be a UFD, so, again, $p$ is not irreducible in ${\cal O}_5$. Let $\pi=r+s{1+\sqrt5\over2}$ be a nontrivial factor of $p$ in ${\cal O}_5$. Taking norms, we get $p=r^2+rs-s^2$ Now a simple substitution gets us to $(a+b)^2+ab$.

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    What about the question I asked you? Namely, what do you mean by "I am searching by computations in di$f$ferent ways"?2012-11-09
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I can answer for your second part. But, you should know about reciprocity. Otherwise, I am sorry. p = 1 (mod 3) is very known. for (-3/p) = (-1/p)(3/p) =) (-1/p) = (-1)^{(p-1)/2} and (3/p) = (-1)^{(p-1)/2}(p/3) =) (-3/p) = (-1)^{(p-1)/2}(-1)^{(p-1)/2}(p/3) = (p/3) i.e., -3 is quadratic residue mod p, if and only of p = 1 (mod 3). NOTE you can try for p = 2 (mod 3) more easily without using reciprocity. I hope you got it.

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    ! why can't we solve with reciprocity?2012-11-09