Lagrange four-squares theorem — deterministic complexity
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Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions in quadratic time. My question is if anything is known about the deterministic time complexity of finding one of the solutions? Any pointers would be appreciated.
(It seems that enumerating all the solutions is hard as factoring in certain cases (via Jacobi's four-square theorem), but correct me if I am wrong.)
nt.number-theory computational-complexity
edited 16 hours ago
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$begingroup$
Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions in quadratic time. My question is if anything is known about the deterministic time complexity of finding one of the solutions? Any pointers would be appreciated.
(It seems that enumerating all the solutions is hard as factoring in certain cases (via Jacobi's four-square theorem), but correct me if I am wrong.)
nt.number-theory computational-complexity
edited 16 hours ago
Tony Huynh
19.8k672131
asked 17 hours ago
Occams_TrimmerOccams_Trimmer
1685
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Occams_Trimmer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1
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Your parenthetical remark is essentially correct. For instance, this is at least as difficult as factoring semiprimes, because it lets us compute $1+p+q+pq$ given a semiprime $pq$, and from $p+q,pq$ it's easy to compute $p,q$.
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– Wojowu
16 hours ago
add a comment |
$begingroup$
Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions in quadratic time. My question is if anything is known about the deterministic time complexity of finding one of the solutions? Any pointers would be appreciated.
(It seems that enumerating all the solutions is hard as factoring in certain cases (via Jacobi's four-square theorem), but correct me if I am wrong.)
nt.number-theory computational-complexity
edited 16 hours ago
Tony Huynh
19.8k672131
asked 17 hours ago
Occams_TrimmerOccams_Trimmer
1685
New contributor
Occams_Trimmer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Lagrange's four-squares theorem states that every natural number can be represented as the sum of four integer squares. Rabin and Shallit gave a randomised algorithm that finds one of these solutions in quadratic time. My question is if anything is known about the deterministic time complexity of finding one of the solutions? Any pointers would be appreciated.
(It seems that enumerating all the solutions is hard as factoring in certain cases (via Jacobi's four-square theorem), but correct me if I am wrong.)
nt.number-theory computational-complexity
nt.number-theory computational-complexity
edited 16 hours ago
Tony Huynh
19.8k672131
asked 17 hours ago
Occams_TrimmerOccams_Trimmer
1685
New contributor
Occams_Trimmer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 16 hours ago
Tony Huynh
19.8k672131
asked 17 hours ago
Occams_TrimmerOccams_Trimmer
1685
New contributor
Occams_Trimmer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 16 hours ago
Tony Huynh
19.8k672131
edited 16 hours ago
Tony Huynh
19.8k672131
edited 16 hours ago
Tony Huynh
19.8k672131
19.8k672131
asked 17 hours ago
Occams_TrimmerOccams_Trimmer
1685
New contributor
Occams_Trimmer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 17 hours ago
Occams_TrimmerOccams_Trimmer
1685
asked 17 hours ago
Occams_TrimmerOccams_Trimmer
1685
1685
New contributor
Occams_Trimmer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Occams_Trimmer is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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1
$begingroup$
Your parenthetical remark is essentially correct. For instance, this is at least as difficult as factoring semiprimes, because it lets us compute $1+p+q+pq$ given a semiprime $pq$, and from $p+q,pq$ it's easy to compute $p,q$.
$endgroup$
– Wojowu
16 hours ago
add a comment |
1
$begingroup$
Your parenthetical remark is essentially correct. For instance, this is at least as difficult as factoring semiprimes, because it lets us compute $1+p+q+pq$ given a semiprime $pq$, and from $p+q,pq$ it's easy to compute $p,q$.
$endgroup$
– Wojowu
16 hours ago
1
1
$begingroup$
Your parenthetical remark is essentially correct. For instance, this is at least as difficult as factoring semiprimes, because it lets us compute $1+p+q+pq$ given a semiprime $pq$, and from $p+q,pq$ it's easy to compute $p,q$.
$endgroup$
– Wojowu
16 hours ago
$begingroup$
Your parenthetical remark is essentially correct. For instance, this is at least as difficult as factoring semiprimes, because it lets us compute $1+p+q+pq$ given a semiprime $pq$, and from $p+q,pq$ it's easy to compute $p,q$.
$endgroup$
– Wojowu
16 hours ago
add a comment |
1 Answer
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As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a deterministic algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time.
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
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$begingroup$
As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a deterministic algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time.
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
$endgroup$
add a comment |
$begingroup$
As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a deterministic algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time.
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
$endgroup$
add a comment |
$begingroup$
As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a deterministic algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time.
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
$endgroup$
As far as I know, this is still an open problem. This is discussed in Section $5$ of the paper Finding the four squares in Lagrange's theorem by Pollack and Treviño. They mention that there is a deterministic algorithm when $n$ is a prime via quaterion multiplication, due to Bumby. Assuming a conjecture of Heath-Brown, there is a deterministic algorithm that works for all $n$. Finally, they mention that a positive proportion of all numbers can be written as the sum of four squares in deterministic polynomial time. Under the Extended Riemann Hypothesis, almost all numbers can be written as the sum of four squares in deterministic polynomial time.
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
answered 16 hours ago
Tony HuynhTony Huynh
19.8k672131
19.8k672131
add a comment |
add a comment |
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$begingroup$
Your parenthetical remark is essentially correct. For instance, this is at least as difficult as factoring semiprimes, because it lets us compute $1+p+q+pq$ given a semiprime $pq$, and from $p+q,pq$ it's easy to compute $p,q$.
$endgroup$
– Wojowu
16 hours ago