Prime number and Divisors
Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?
I came across this question in a Math Olympiad Competition and had no idea how to solve it
elementary-number-theory prime-numbers
edited 29 mins ago
Chinnapparaj R
5,2271826
asked 48 mins ago
Mohammad Mizanur Rahaman
85
add a comment |
Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?
I came across this question in a Math Olympiad Competition and had no idea how to solve it
elementary-number-theory prime-numbers
edited 29 mins ago
Chinnapparaj R
5,2271826
asked 48 mins ago
Mohammad Mizanur Rahaman
85
4
A number with an odd number of divisors must be square.
– Lord Shark the Unknown
46 mins ago
add a comment |
Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?
I came across this question in a Math Olympiad Competition and had no idea how to solve it
elementary-number-theory prime-numbers
edited 29 mins ago
Chinnapparaj R
5,2271826
asked 48 mins ago
Mohammad Mizanur Rahaman
85
Let $p$ be a prime number such that $p^2+12$ has exactly $5$ divisors. What is the maximum value of $p$ ?
I came across this question in a Math Olympiad Competition and had no idea how to solve it
elementary-number-theory prime-numbers
elementary-number-theory prime-numbers
edited 29 mins ago
Chinnapparaj R
5,2271826
asked 48 mins ago
Mohammad Mizanur Rahaman
85
edited 29 mins ago
Chinnapparaj R
5,2271826
asked 48 mins ago
Mohammad Mizanur Rahaman
85
edited 29 mins ago
Chinnapparaj R
5,2271826
edited 29 mins ago
Chinnapparaj R
5,2271826
edited 29 mins ago
Chinnapparaj R
5,2271826
5,2271826
asked 48 mins ago
Mohammad Mizanur Rahaman
85
asked 48 mins ago
Mohammad Mizanur Rahaman
85
asked 48 mins ago
Mohammad Mizanur Rahaman
85
85
4
A number with an odd number of divisors must be square.
– Lord Shark the Unknown
46 mins ago
add a comment |
4
A number with an odd number of divisors must be square.
– Lord Shark the Unknown
46 mins ago
4
4
A number with an odd number of divisors must be square.
– Lord Shark the Unknown
46 mins ago
A number with an odd number of divisors must be square.
– Lord Shark the Unknown
46 mins ago
add a comment |
1 Answer
1
active
oldest
votes
Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:
$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$
Then do some casework on it.
answered 43 mins ago
Barycentric_Bash
37728
The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
35 mins ago
1
@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
24 mins ago
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:
$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$
Then do some casework on it.
answered 43 mins ago
Barycentric_Bash
37728
The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
35 mins ago
1
@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
24 mins ago
add a comment |
Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:
$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$
Then do some casework on it.
answered 43 mins ago
Barycentric_Bash
37728
The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
35 mins ago
1
@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
24 mins ago
add a comment |
Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:
$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$
Then do some casework on it.
answered 43 mins ago
Barycentric_Bash
37728
Notice that a number that has 5 divisors must be in the form of $q^{4}$ for some prime $q$. So we get:
$q^{4}=p^{2}+12 implies (q^{2}-p)(q^{2}+p)=12$
Then do some casework on it.
answered 43 mins ago
Barycentric_Bash
37728
answered 43 mins ago
Barycentric_Bash
37728
answered 43 mins ago
Barycentric_Bash
37728
answered 43 mins ago
Barycentric_Bash
37728
37728
The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
35 mins ago
1
@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
24 mins ago
add a comment |
The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
35 mins ago
1
@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
24 mins ago
The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
35 mins ago
The answer I have got is 2(maximum value of P). I think I am right ?
– Mohammad Mizanur Rahaman
35 mins ago
1
1
@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
24 mins ago
@MohammadMizanurRahaman: yes, $p=q=2$
– Ross Millikan
24 mins ago
add a comment |
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4
A number with an odd number of divisors must be square.
– Lord Shark the Unknown
46 mins ago