Show that $2n^2+11$ and $2n^2+29$ generate primes for all non-negative integers $n<11$ and $n<29$...
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Show that $2n^2+11$ and $2n^2+29$ generate primes for all non-negative integers $n<11$ and $n<29$ respectively.
This question is from Elementary Number Theory by Kenneth Rosen. The book's solution has used a brute force method to generate all numbers and manually check if they are prime or not.
I wish to ask instead if there is a better method. I have looked around on the internet and couldn't find anything. I instead wrote a C program to test the theory about primes of the form $2n^2+x$ $(0leq n<x)$, and it seems only $x=3,5,11,29$ satisfy it under $10^6$.
So, is there no better, more number theoretical way to verify the given question?
elementary-number-theory
add a comment |
up vote
3
down vote
favorite
Show that $2n^2+11$ and $2n^2+29$ generate primes for all non-negative integers $n<11$ and $n<29$ respectively.
This question is from Elementary Number Theory by Kenneth Rosen. The book's solution has used a brute force method to generate all numbers and manually check if they are prime or not.
I wish to ask instead if there is a better method. I have looked around on the internet and couldn't find anything. I instead wrote a C program to test the theory about primes of the form $2n^2+x$ $(0leq n<x)$, and it seems only $x=3,5,11,29$ satisfy it under $10^6$.
So, is there no better, more number theoretical way to verify the given question?
elementary-number-theory
there might be a sensible proof, not sure yet. Here is the proof for the better known $n^2 + n + 41$ math.stackexchange.com/questions/289338/…
– Will Jagy
Nov 14 at 2:46
For $2n^2+11 leq 2cdot 11^2 + 11 = 253$ and hence it is enough to check that $2n^2+11 mod p neq 0$ for $p < sqrt{253} $. Thus we need to check for $p=2,3,5,7,11,13$. This is not hard.
– Muralidharan
Nov 14 at 2:53
ُ$x^2+x+17$, $2x^2+29$, $x^2+x+41$ and $x^2-29x+160$ are Euler numbers which give primes for 0=<n<x
– sirous
yesterday
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Show that $2n^2+11$ and $2n^2+29$ generate primes for all non-negative integers $n<11$ and $n<29$ respectively.
This question is from Elementary Number Theory by Kenneth Rosen. The book's solution has used a brute force method to generate all numbers and manually check if they are prime or not.
I wish to ask instead if there is a better method. I have looked around on the internet and couldn't find anything. I instead wrote a C program to test the theory about primes of the form $2n^2+x$ $(0leq n<x)$, and it seems only $x=3,5,11,29$ satisfy it under $10^6$.
So, is there no better, more number theoretical way to verify the given question?
elementary-number-theory
Show that $2n^2+11$ and $2n^2+29$ generate primes for all non-negative integers $n<11$ and $n<29$ respectively.
This question is from Elementary Number Theory by Kenneth Rosen. The book's solution has used a brute force method to generate all numbers and manually check if they are prime or not.
I wish to ask instead if there is a better method. I have looked around on the internet and couldn't find anything. I instead wrote a C program to test the theory about primes of the form $2n^2+x$ $(0leq n<x)$, and it seems only $x=3,5,11,29$ satisfy it under $10^6$.
So, is there no better, more number theoretical way to verify the given question?
elementary-number-theory
elementary-number-theory
asked Nov 14 at 2:39
Gaurang Tandon
3,48622147
3,48622147
there might be a sensible proof, not sure yet. Here is the proof for the better known $n^2 + n + 41$ math.stackexchange.com/questions/289338/…
– Will Jagy
Nov 14 at 2:46
For $2n^2+11 leq 2cdot 11^2 + 11 = 253$ and hence it is enough to check that $2n^2+11 mod p neq 0$ for $p < sqrt{253} $. Thus we need to check for $p=2,3,5,7,11,13$. This is not hard.
– Muralidharan
Nov 14 at 2:53
ُ$x^2+x+17$, $2x^2+29$, $x^2+x+41$ and $x^2-29x+160$ are Euler numbers which give primes for 0=<n<x
– sirous
yesterday
add a comment |
there might be a sensible proof, not sure yet. Here is the proof for the better known $n^2 + n + 41$ math.stackexchange.com/questions/289338/…
– Will Jagy
Nov 14 at 2:46
For $2n^2+11 leq 2cdot 11^2 + 11 = 253$ and hence it is enough to check that $2n^2+11 mod p neq 0$ for $p < sqrt{253} $. Thus we need to check for $p=2,3,5,7,11,13$. This is not hard.
– Muralidharan
Nov 14 at 2:53
ُ$x^2+x+17$, $2x^2+29$, $x^2+x+41$ and $x^2-29x+160$ are Euler numbers which give primes for 0=<n<x
– sirous
yesterday
there might be a sensible proof, not sure yet. Here is the proof for the better known $n^2 + n + 41$ math.stackexchange.com/questions/289338/…
– Will Jagy
Nov 14 at 2:46
there might be a sensible proof, not sure yet. Here is the proof for the better known $n^2 + n + 41$ math.stackexchange.com/questions/289338/…
– Will Jagy
Nov 14 at 2:46
For $2n^2+11 leq 2cdot 11^2 + 11 = 253$ and hence it is enough to check that $2n^2+11 mod p neq 0$ for $p < sqrt{253} $. Thus we need to check for $p=2,3,5,7,11,13$. This is not hard.
– Muralidharan
Nov 14 at 2:53
For $2n^2+11 leq 2cdot 11^2 + 11 = 253$ and hence it is enough to check that $2n^2+11 mod p neq 0$ for $p < sqrt{253} $. Thus we need to check for $p=2,3,5,7,11,13$. This is not hard.
– Muralidharan
Nov 14 at 2:53
ُ$x^2+x+17$, $2x^2+29$, $x^2+x+41$ and $x^2-29x+160$ are Euler numbers which give primes for 0=<n<x
– sirous
yesterday
ُ$x^2+x+17$, $2x^2+29$, $x^2+x+41$ and $x^2-29x+160$ are Euler numbers which give primes for 0=<n<x
– sirous
yesterday
add a comment |
2 Answers
2
active
oldest
votes
up vote
3
down vote
accepted
Context: here is a proof for the better-known $n^2 + n + 41$ Is the notorious $n^2 + n + 41$ prime generator the last of its type?
Took me a while and some numerical experiments, but I got this one. For foundational material on positive binary quadratic forms, see https://en.wikipedia.org/wiki/Binary_quadratic_form and https://en.wikipedia.org/wiki/Binary_quadratic_form#References My favorite book is Buell; people seem to like Cox.
We are given a prime $p equiv pm 3 pmod 8$ such that the form class number $h(-8p) = 2.$ In particular, the only classes are those of $$ x^2 + 2py^2 $$ and $$ 2 x^2 + p y^2. $$ The shorthand for these two forms is $langle 1,0,2p rangle$ and $langle 2,0,p rangle ; . ;$
Note that the product $2p$ must be one of Euler's Idoneal Numbers
We will ASSUME that there is an integer $n$ with $1 leq n < p$ such that $2n^2 + p$ is composite. We will show that this leads o a contradiction by constructing a third primitive quadratic form of the same discriminant.
Let $q$ be the smallest prime that divides any $2n^2 + p$ with $n < p.$ Usually there will be several small $n$ giving the same $q.$ In turn, take the smallest $n$ that works, meaning $2n^2 + p equiv 0 pmod q ; . ;$ We get $q$ prime,
$$ 2n^2 + p = q t ; , ; $$
$$ t > q > 2n ; . ; $$
There are just two cases.
IF $q > 4 n,$ then
$$ langle q,4n,2t rangle $$
is a reduced primitive form of the same discriminant $(-8p).$ It is not equivalent to either of the original forms listed, that is what we get for reduced forms.
IF $q < 4 n,$ then
$$ langle q,4n-2q, 2t - 4n + q rangle $$
is a reduced primitive form of the same discriminant $(-8p).$
In either case, the presence of composite numbers represented as $2n^2 + p$ has resulted in $h(-8p) geq 3,$ contradicting the hypothesis of class number 2.
Examples for the first case:
If $p equiv 3 pmod 5,$ we can take $n=1$ and get third form
$$ langle 5,4, frac{2p+4}{5} rangle $$
If $p equiv 3 pmod {11},$ we can take $n=2$ and get third form
$$ langle 11,8, frac{2p+16}{11} rangle $$
Examples for the second case:
If $p equiv 1 pmod 3,$ we can take $n=1$ and get third form
$$ langle 3,2, frac{2p+1}{3} rangle $$
If $p equiv 2 pmod 5,$ we can take $n=2$ and get third form
$$ langle 5,2, frac{2p+1}{5} rangle $$
add a comment |
up vote
1
down vote
Alright, I will need to work on this a bit, but it is already clear that your condition is that (binary quadratic) form class number $h(-8p) = 2$ for prime $p.$ Note that your $p equiv pm 3 pmod 8,$ otherwise the form $langle 2,0,p rangle$ would be in the principal genus, but the presence of two ambiguous forms demands a second genus, therefore the class number would be at least four.
Discr -24 = 2^3 * 3 class number 2
all
24: < 1, 0, 6> Square 24: < 1, 0, 6>
24: < 2, 0, 3> Square 24: < 1, 0, 6>
Discr -40 = 2^3 * 5 class number 2
all
40: < 1, 0, 10> Square 40: < 1, 0, 10>
40: < 2, 0, 5> Square 40: < 1, 0, 10>
Discr -88 = 2^3 * 11 class number 2
all
88: < 1, 0, 22> Square 88: < 1, 0, 22>
88: < 2, 0, 11> Square 88: < 1, 0, 22>
Discr -232 = 2^3 * 29 class number 2
all
232: < 1, 0, 58> Square 232: < 1, 0, 58>
232: < 2, 0, 29> Square 232: < 1, 0, 58>
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Context: here is a proof for the better-known $n^2 + n + 41$ Is the notorious $n^2 + n + 41$ prime generator the last of its type?
Took me a while and some numerical experiments, but I got this one. For foundational material on positive binary quadratic forms, see https://en.wikipedia.org/wiki/Binary_quadratic_form and https://en.wikipedia.org/wiki/Binary_quadratic_form#References My favorite book is Buell; people seem to like Cox.
We are given a prime $p equiv pm 3 pmod 8$ such that the form class number $h(-8p) = 2.$ In particular, the only classes are those of $$ x^2 + 2py^2 $$ and $$ 2 x^2 + p y^2. $$ The shorthand for these two forms is $langle 1,0,2p rangle$ and $langle 2,0,p rangle ; . ;$
Note that the product $2p$ must be one of Euler's Idoneal Numbers
We will ASSUME that there is an integer $n$ with $1 leq n < p$ such that $2n^2 + p$ is composite. We will show that this leads o a contradiction by constructing a third primitive quadratic form of the same discriminant.
Let $q$ be the smallest prime that divides any $2n^2 + p$ with $n < p.$ Usually there will be several small $n$ giving the same $q.$ In turn, take the smallest $n$ that works, meaning $2n^2 + p equiv 0 pmod q ; . ;$ We get $q$ prime,
$$ 2n^2 + p = q t ; , ; $$
$$ t > q > 2n ; . ; $$
There are just two cases.
IF $q > 4 n,$ then
$$ langle q,4n,2t rangle $$
is a reduced primitive form of the same discriminant $(-8p).$ It is not equivalent to either of the original forms listed, that is what we get for reduced forms.
IF $q < 4 n,$ then
$$ langle q,4n-2q, 2t - 4n + q rangle $$
is a reduced primitive form of the same discriminant $(-8p).$
In either case, the presence of composite numbers represented as $2n^2 + p$ has resulted in $h(-8p) geq 3,$ contradicting the hypothesis of class number 2.
Examples for the first case:
If $p equiv 3 pmod 5,$ we can take $n=1$ and get third form
$$ langle 5,4, frac{2p+4}{5} rangle $$
If $p equiv 3 pmod {11},$ we can take $n=2$ and get third form
$$ langle 11,8, frac{2p+16}{11} rangle $$
Examples for the second case:
If $p equiv 1 pmod 3,$ we can take $n=1$ and get third form
$$ langle 3,2, frac{2p+1}{3} rangle $$
If $p equiv 2 pmod 5,$ we can take $n=2$ and get third form
$$ langle 5,2, frac{2p+1}{5} rangle $$
add a comment |
up vote
3
down vote
accepted
Context: here is a proof for the better-known $n^2 + n + 41$ Is the notorious $n^2 + n + 41$ prime generator the last of its type?
Took me a while and some numerical experiments, but I got this one. For foundational material on positive binary quadratic forms, see https://en.wikipedia.org/wiki/Binary_quadratic_form and https://en.wikipedia.org/wiki/Binary_quadratic_form#References My favorite book is Buell; people seem to like Cox.
We are given a prime $p equiv pm 3 pmod 8$ such that the form class number $h(-8p) = 2.$ In particular, the only classes are those of $$ x^2 + 2py^2 $$ and $$ 2 x^2 + p y^2. $$ The shorthand for these two forms is $langle 1,0,2p rangle$ and $langle 2,0,p rangle ; . ;$
Note that the product $2p$ must be one of Euler's Idoneal Numbers
We will ASSUME that there is an integer $n$ with $1 leq n < p$ such that $2n^2 + p$ is composite. We will show that this leads o a contradiction by constructing a third primitive quadratic form of the same discriminant.
Let $q$ be the smallest prime that divides any $2n^2 + p$ with $n < p.$ Usually there will be several small $n$ giving the same $q.$ In turn, take the smallest $n$ that works, meaning $2n^2 + p equiv 0 pmod q ; . ;$ We get $q$ prime,
$$ 2n^2 + p = q t ; , ; $$
$$ t > q > 2n ; . ; $$
There are just two cases.
IF $q > 4 n,$ then
$$ langle q,4n,2t rangle $$
is a reduced primitive form of the same discriminant $(-8p).$ It is not equivalent to either of the original forms listed, that is what we get for reduced forms.
IF $q < 4 n,$ then
$$ langle q,4n-2q, 2t - 4n + q rangle $$
is a reduced primitive form of the same discriminant $(-8p).$
In either case, the presence of composite numbers represented as $2n^2 + p$ has resulted in $h(-8p) geq 3,$ contradicting the hypothesis of class number 2.
Examples for the first case:
If $p equiv 3 pmod 5,$ we can take $n=1$ and get third form
$$ langle 5,4, frac{2p+4}{5} rangle $$
If $p equiv 3 pmod {11},$ we can take $n=2$ and get third form
$$ langle 11,8, frac{2p+16}{11} rangle $$
Examples for the second case:
If $p equiv 1 pmod 3,$ we can take $n=1$ and get third form
$$ langle 3,2, frac{2p+1}{3} rangle $$
If $p equiv 2 pmod 5,$ we can take $n=2$ and get third form
$$ langle 5,2, frac{2p+1}{5} rangle $$
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Context: here is a proof for the better-known $n^2 + n + 41$ Is the notorious $n^2 + n + 41$ prime generator the last of its type?
Took me a while and some numerical experiments, but I got this one. For foundational material on positive binary quadratic forms, see https://en.wikipedia.org/wiki/Binary_quadratic_form and https://en.wikipedia.org/wiki/Binary_quadratic_form#References My favorite book is Buell; people seem to like Cox.
We are given a prime $p equiv pm 3 pmod 8$ such that the form class number $h(-8p) = 2.$ In particular, the only classes are those of $$ x^2 + 2py^2 $$ and $$ 2 x^2 + p y^2. $$ The shorthand for these two forms is $langle 1,0,2p rangle$ and $langle 2,0,p rangle ; . ;$
Note that the product $2p$ must be one of Euler's Idoneal Numbers
We will ASSUME that there is an integer $n$ with $1 leq n < p$ such that $2n^2 + p$ is composite. We will show that this leads o a contradiction by constructing a third primitive quadratic form of the same discriminant.
Let $q$ be the smallest prime that divides any $2n^2 + p$ with $n < p.$ Usually there will be several small $n$ giving the same $q.$ In turn, take the smallest $n$ that works, meaning $2n^2 + p equiv 0 pmod q ; . ;$ We get $q$ prime,
$$ 2n^2 + p = q t ; , ; $$
$$ t > q > 2n ; . ; $$
There are just two cases.
IF $q > 4 n,$ then
$$ langle q,4n,2t rangle $$
is a reduced primitive form of the same discriminant $(-8p).$ It is not equivalent to either of the original forms listed, that is what we get for reduced forms.
IF $q < 4 n,$ then
$$ langle q,4n-2q, 2t - 4n + q rangle $$
is a reduced primitive form of the same discriminant $(-8p).$
In either case, the presence of composite numbers represented as $2n^2 + p$ has resulted in $h(-8p) geq 3,$ contradicting the hypothesis of class number 2.
Examples for the first case:
If $p equiv 3 pmod 5,$ we can take $n=1$ and get third form
$$ langle 5,4, frac{2p+4}{5} rangle $$
If $p equiv 3 pmod {11},$ we can take $n=2$ and get third form
$$ langle 11,8, frac{2p+16}{11} rangle $$
Examples for the second case:
If $p equiv 1 pmod 3,$ we can take $n=1$ and get third form
$$ langle 3,2, frac{2p+1}{3} rangle $$
If $p equiv 2 pmod 5,$ we can take $n=2$ and get third form
$$ langle 5,2, frac{2p+1}{5} rangle $$
Context: here is a proof for the better-known $n^2 + n + 41$ Is the notorious $n^2 + n + 41$ prime generator the last of its type?
Took me a while and some numerical experiments, but I got this one. For foundational material on positive binary quadratic forms, see https://en.wikipedia.org/wiki/Binary_quadratic_form and https://en.wikipedia.org/wiki/Binary_quadratic_form#References My favorite book is Buell; people seem to like Cox.
We are given a prime $p equiv pm 3 pmod 8$ such that the form class number $h(-8p) = 2.$ In particular, the only classes are those of $$ x^2 + 2py^2 $$ and $$ 2 x^2 + p y^2. $$ The shorthand for these two forms is $langle 1,0,2p rangle$ and $langle 2,0,p rangle ; . ;$
Note that the product $2p$ must be one of Euler's Idoneal Numbers
We will ASSUME that there is an integer $n$ with $1 leq n < p$ such that $2n^2 + p$ is composite. We will show that this leads o a contradiction by constructing a third primitive quadratic form of the same discriminant.
Let $q$ be the smallest prime that divides any $2n^2 + p$ with $n < p.$ Usually there will be several small $n$ giving the same $q.$ In turn, take the smallest $n$ that works, meaning $2n^2 + p equiv 0 pmod q ; . ;$ We get $q$ prime,
$$ 2n^2 + p = q t ; , ; $$
$$ t > q > 2n ; . ; $$
There are just two cases.
IF $q > 4 n,$ then
$$ langle q,4n,2t rangle $$
is a reduced primitive form of the same discriminant $(-8p).$ It is not equivalent to either of the original forms listed, that is what we get for reduced forms.
IF $q < 4 n,$ then
$$ langle q,4n-2q, 2t - 4n + q rangle $$
is a reduced primitive form of the same discriminant $(-8p).$
In either case, the presence of composite numbers represented as $2n^2 + p$ has resulted in $h(-8p) geq 3,$ contradicting the hypothesis of class number 2.
Examples for the first case:
If $p equiv 3 pmod 5,$ we can take $n=1$ and get third form
$$ langle 5,4, frac{2p+4}{5} rangle $$
If $p equiv 3 pmod {11},$ we can take $n=2$ and get third form
$$ langle 11,8, frac{2p+16}{11} rangle $$
Examples for the second case:
If $p equiv 1 pmod 3,$ we can take $n=1$ and get third form
$$ langle 3,2, frac{2p+1}{3} rangle $$
If $p equiv 2 pmod 5,$ we can take $n=2$ and get third form
$$ langle 5,2, frac{2p+1}{5} rangle $$
edited 2 days ago
answered 2 days ago
Will Jagy
100k597198
100k597198
add a comment |
add a comment |
up vote
1
down vote
Alright, I will need to work on this a bit, but it is already clear that your condition is that (binary quadratic) form class number $h(-8p) = 2$ for prime $p.$ Note that your $p equiv pm 3 pmod 8,$ otherwise the form $langle 2,0,p rangle$ would be in the principal genus, but the presence of two ambiguous forms demands a second genus, therefore the class number would be at least four.
Discr -24 = 2^3 * 3 class number 2
all
24: < 1, 0, 6> Square 24: < 1, 0, 6>
24: < 2, 0, 3> Square 24: < 1, 0, 6>
Discr -40 = 2^3 * 5 class number 2
all
40: < 1, 0, 10> Square 40: < 1, 0, 10>
40: < 2, 0, 5> Square 40: < 1, 0, 10>
Discr -88 = 2^3 * 11 class number 2
all
88: < 1, 0, 22> Square 88: < 1, 0, 22>
88: < 2, 0, 11> Square 88: < 1, 0, 22>
Discr -232 = 2^3 * 29 class number 2
all
232: < 1, 0, 58> Square 232: < 1, 0, 58>
232: < 2, 0, 29> Square 232: < 1, 0, 58>
add a comment |
up vote
1
down vote
Alright, I will need to work on this a bit, but it is already clear that your condition is that (binary quadratic) form class number $h(-8p) = 2$ for prime $p.$ Note that your $p equiv pm 3 pmod 8,$ otherwise the form $langle 2,0,p rangle$ would be in the principal genus, but the presence of two ambiguous forms demands a second genus, therefore the class number would be at least four.
Discr -24 = 2^3 * 3 class number 2
all
24: < 1, 0, 6> Square 24: < 1, 0, 6>
24: < 2, 0, 3> Square 24: < 1, 0, 6>
Discr -40 = 2^3 * 5 class number 2
all
40: < 1, 0, 10> Square 40: < 1, 0, 10>
40: < 2, 0, 5> Square 40: < 1, 0, 10>
Discr -88 = 2^3 * 11 class number 2
all
88: < 1, 0, 22> Square 88: < 1, 0, 22>
88: < 2, 0, 11> Square 88: < 1, 0, 22>
Discr -232 = 2^3 * 29 class number 2
all
232: < 1, 0, 58> Square 232: < 1, 0, 58>
232: < 2, 0, 29> Square 232: < 1, 0, 58>
add a comment |
up vote
1
down vote
up vote
1
down vote
Alright, I will need to work on this a bit, but it is already clear that your condition is that (binary quadratic) form class number $h(-8p) = 2$ for prime $p.$ Note that your $p equiv pm 3 pmod 8,$ otherwise the form $langle 2,0,p rangle$ would be in the principal genus, but the presence of two ambiguous forms demands a second genus, therefore the class number would be at least four.
Discr -24 = 2^3 * 3 class number 2
all
24: < 1, 0, 6> Square 24: < 1, 0, 6>
24: < 2, 0, 3> Square 24: < 1, 0, 6>
Discr -40 = 2^3 * 5 class number 2
all
40: < 1, 0, 10> Square 40: < 1, 0, 10>
40: < 2, 0, 5> Square 40: < 1, 0, 10>
Discr -88 = 2^3 * 11 class number 2
all
88: < 1, 0, 22> Square 88: < 1, 0, 22>
88: < 2, 0, 11> Square 88: < 1, 0, 22>
Discr -232 = 2^3 * 29 class number 2
all
232: < 1, 0, 58> Square 232: < 1, 0, 58>
232: < 2, 0, 29> Square 232: < 1, 0, 58>
Alright, I will need to work on this a bit, but it is already clear that your condition is that (binary quadratic) form class number $h(-8p) = 2$ for prime $p.$ Note that your $p equiv pm 3 pmod 8,$ otherwise the form $langle 2,0,p rangle$ would be in the principal genus, but the presence of two ambiguous forms demands a second genus, therefore the class number would be at least four.
Discr -24 = 2^3 * 3 class number 2
all
24: < 1, 0, 6> Square 24: < 1, 0, 6>
24: < 2, 0, 3> Square 24: < 1, 0, 6>
Discr -40 = 2^3 * 5 class number 2
all
40: < 1, 0, 10> Square 40: < 1, 0, 10>
40: < 2, 0, 5> Square 40: < 1, 0, 10>
Discr -88 = 2^3 * 11 class number 2
all
88: < 1, 0, 22> Square 88: < 1, 0, 22>
88: < 2, 0, 11> Square 88: < 1, 0, 22>
Discr -232 = 2^3 * 29 class number 2
all
232: < 1, 0, 58> Square 232: < 1, 0, 58>
232: < 2, 0, 29> Square 232: < 1, 0, 58>
edited Nov 14 at 3:00
answered Nov 14 at 2:52
Will Jagy
100k597198
100k597198
add a comment |
add a comment |
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there might be a sensible proof, not sure yet. Here is the proof for the better known $n^2 + n + 41$ math.stackexchange.com/questions/289338/…
– Will Jagy
Nov 14 at 2:46
For $2n^2+11 leq 2cdot 11^2 + 11 = 253$ and hence it is enough to check that $2n^2+11 mod p neq 0$ for $p < sqrt{253} $. Thus we need to check for $p=2,3,5,7,11,13$. This is not hard.
– Muralidharan
Nov 14 at 2:53
ُ$x^2+x+17$, $2x^2+29$, $x^2+x+41$ and $x^2-29x+160$ are Euler numbers which give primes for 0=<n<x
– sirous
yesterday