Primes of the form $p=x^4+y^4$
$begingroup$
Are there infinitely many prime numbers $p$ such that
$$p = x^4+y^4$$
for some $x,y in Bbb Z$ ? What if we only require $x,y in Bbb Q$ ?
I know that $p = a^2+b^2$ with $a,b in Bbb Q$ iff $p = a^2+b^2$ with $a,b in Bbb Z$ (Davenport-Cassels) iff $p=2$ or $p equiv 1 pmod 4$. But $a$ and $b$ might not be squares. What are some references about this problem, which explain what is (un)known?
number-theory polynomials prime-numbers
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
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|
show 4 more comments
$begingroup$
Are there infinitely many prime numbers $p$ such that
$$p = x^4+y^4$$
for some $x,y in Bbb Z$ ? What if we only require $x,y in Bbb Q$ ?
I know that $p = a^2+b^2$ with $a,b in Bbb Q$ iff $p = a^2+b^2$ with $a,b in Bbb Z$ (Davenport-Cassels) iff $p=2$ or $p equiv 1 pmod 4$. But $a$ and $b$ might not be squares. What are some references about this problem, which explain what is (un)known?
number-theory polynomials prime-numbers
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
$endgroup$
3
$begingroup$
They are called quartan primes. See A002645. I didn't find a proof in the references though.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 9:56
$begingroup$
Also, what about $x^3+y^3$ and $x^5+y^5$ ?
$endgroup$
– Alphonse
Dec 1 '18 at 10:12
3
$begingroup$
@Alphonse Divisible by $x+y$.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 10:18
5
$begingroup$
Notice that it was proved only in 1998 that there are infinitely many primes of the form $a^2+y^4$.
$endgroup$
– Watson
Dec 1 '18 at 16:32
2
$begingroup$
Yes, one expects infinitely many primes of the form $3x^2-3x+1$ for $x$ integral. More precise versions of this follow from a conjecture of Bateman and Horn.
$endgroup$
– Mike Bennett
Dec 2 '18 at 17:14
|
show 4 more comments
$begingroup$
Are there infinitely many prime numbers $p$ such that
$$p = x^4+y^4$$
for some $x,y in Bbb Z$ ? What if we only require $x,y in Bbb Q$ ?
I know that $p = a^2+b^2$ with $a,b in Bbb Q$ iff $p = a^2+b^2$ with $a,b in Bbb Z$ (Davenport-Cassels) iff $p=2$ or $p equiv 1 pmod 4$. But $a$ and $b$ might not be squares. What are some references about this problem, which explain what is (un)known?
number-theory polynomials prime-numbers
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
$endgroup$
Are there infinitely many prime numbers $p$ such that
$$p = x^4+y^4$$
for some $x,y in Bbb Z$ ? What if we only require $x,y in Bbb Q$ ?
I know that $p = a^2+b^2$ with $a,b in Bbb Q$ iff $p = a^2+b^2$ with $a,b in Bbb Z$ (Davenport-Cassels) iff $p=2$ or $p equiv 1 pmod 4$. But $a$ and $b$ might not be squares. What are some references about this problem, which explain what is (un)known?
number-theory polynomials prime-numbers
number-theory polynomials prime-numbers
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
edited Dec 1 '18 at 9:50
Alphonse
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
asked Dec 1 '18 at 9:41
AlphonseAlphonse
2,198624
2,198624
3
$begingroup$
They are called quartan primes. See A002645. I didn't find a proof in the references though.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 9:56
$begingroup$
Also, what about $x^3+y^3$ and $x^5+y^5$ ?
$endgroup$
– Alphonse
Dec 1 '18 at 10:12
3
$begingroup$
@Alphonse Divisible by $x+y$.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 10:18
5
$begingroup$
Notice that it was proved only in 1998 that there are infinitely many primes of the form $a^2+y^4$.
$endgroup$
– Watson
Dec 1 '18 at 16:32
2
$begingroup$
Yes, one expects infinitely many primes of the form $3x^2-3x+1$ for $x$ integral. More precise versions of this follow from a conjecture of Bateman and Horn.
$endgroup$
– Mike Bennett
Dec 2 '18 at 17:14
|
show 4 more comments
3
$begingroup$
They are called quartan primes. See A002645. I didn't find a proof in the references though.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 9:56
$begingroup$
Also, what about $x^3+y^3$ and $x^5+y^5$ ?
$endgroup$
– Alphonse
Dec 1 '18 at 10:12
3
$begingroup$
@Alphonse Divisible by $x+y$.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 10:18
5
$begingroup$
Notice that it was proved only in 1998 that there are infinitely many primes of the form $a^2+y^4$.
$endgroup$
– Watson
Dec 1 '18 at 16:32
2
$begingroup$
Yes, one expects infinitely many primes of the form $3x^2-3x+1$ for $x$ integral. More precise versions of this follow from a conjecture of Bateman and Horn.
$endgroup$
– Mike Bennett
Dec 2 '18 at 17:14
3
3
$begingroup$
They are called quartan primes. See A002645. I didn't find a proof in the references though.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 9:56
$begingroup$
They are called quartan primes. See A002645. I didn't find a proof in the references though.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 9:56
$begingroup$
Also, what about $x^3+y^3$ and $x^5+y^5$ ?
$endgroup$
– Alphonse
Dec 1 '18 at 10:12
$begingroup$
Also, what about $x^3+y^3$ and $x^5+y^5$ ?
$endgroup$
– Alphonse
Dec 1 '18 at 10:12
3
3
$begingroup$
@Alphonse Divisible by $x+y$.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 10:18
$begingroup$
@Alphonse Divisible by $x+y$.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 10:18
5
5
$begingroup$
Notice that it was proved only in 1998 that there are infinitely many primes of the form $a^2+y^4$.
$endgroup$
– Watson
Dec 1 '18 at 16:32
$begingroup$
Notice that it was proved only in 1998 that there are infinitely many primes of the form $a^2+y^4$.
$endgroup$
– Watson
Dec 1 '18 at 16:32
2
2
$begingroup$
Yes, one expects infinitely many primes of the form $3x^2-3x+1$ for $x$ integral. More precise versions of this follow from a conjecture of Bateman and Horn.
$endgroup$
– Mike Bennett
Dec 2 '18 at 17:14
$begingroup$
Yes, one expects infinitely many primes of the form $3x^2-3x+1$ for $x$ integral. More precise versions of this follow from a conjecture of Bateman and Horn.
$endgroup$
– Mike Bennett
Dec 2 '18 at 17:14
|
show 4 more comments
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3
$begingroup$
They are called quartan primes. See A002645. I didn't find a proof in the references though.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 9:56
$begingroup$
Also, what about $x^3+y^3$ and $x^5+y^5$ ?
$endgroup$
– Alphonse
Dec 1 '18 at 10:12
3
$begingroup$
@Alphonse Divisible by $x+y$.
$endgroup$
– Jean-Claude Arbaut
Dec 1 '18 at 10:18
5
$begingroup$
Notice that it was proved only in 1998 that there are infinitely many primes of the form $a^2+y^4$.
$endgroup$
– Watson
Dec 1 '18 at 16:32
2
$begingroup$
Yes, one expects infinitely many primes of the form $3x^2-3x+1$ for $x$ integral. More precise versions of this follow from a conjecture of Bateman and Horn.
$endgroup$
– Mike Bennett
Dec 2 '18 at 17:14