Literature Review for $a^5 + b^5 = c^5 + d^5$ [closed]
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I am preparing a paper on a^5 + b^5 = c^5 + d^5, and am having considerable difficulty in the literature review.
I don't see anything on StackExchange.
According to Wolfram's Mathworld, a book (Guy, 1994, page 140) includes the assertion that this equation has been checked to 10^26 with no solution.
(http://mathworld.wolfram.com/DiophantineEquation5thPowers.html) However, I don't have a copy of that book and it isn't free online.
I was a little surprised that 1994 is the most current reference on this problem that has been studied since at least 300 - 400 years ago.
Q: Is there any information available on this problem?
diophantine-equations
asked Dec 31 '18 at 1:08
L1-AL1-A
365
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closed as too broad by Eevee Trainer, Lord Shark the Unknown, Holo, José Carlos Santos, egreg Dec 31 '18 at 14:05
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 4 more comments
$begingroup$
I am preparing a paper on a^5 + b^5 = c^5 + d^5, and am having considerable difficulty in the literature review.
I don't see anything on StackExchange.
According to Wolfram's Mathworld, a book (Guy, 1994, page 140) includes the assertion that this equation has been checked to 10^26 with no solution.
(http://mathworld.wolfram.com/DiophantineEquation5thPowers.html) However, I don't have a copy of that book and it isn't free online.
I was a little surprised that 1994 is the most current reference on this problem that has been studied since at least 300 - 400 years ago.
Q: Is there any information available on this problem?
diophantine-equations
asked Dec 31 '18 at 1:08
L1-AL1-A
365
$endgroup$
closed as too broad by Eevee Trainer, Lord Shark the Unknown, Holo, José Carlos Santos, egreg Dec 31 '18 at 14:05
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/…
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– Will Jagy
Dec 31 '18 at 1:19
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I see. Table on page 453, it says for problem abbreviated 5.2.2 there are no known solutions, checked up to 2.0 * 10^14. As that paper was 1967, there could be substantially bigger searches in the intervening 50 years. It appears that your best bet is Andrew Bremner (1981) in the Journal of Number Theory, but it is clear he found no 5.2.2 or Guy and yur site would have said so.
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– Will Jagy
Dec 31 '18 at 1:28
1
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the book by Guy: books.google.com/… THE BAD NEWS is that it does not show the pages you want. Of course, those pages just report the stuff in those articles
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– Will Jagy
Dec 31 '18 at 2:02
1
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Mathworld cites the same result Guy cites, which I take as evidence that no one has checked further. mathworld.wolfram.com/DiophantineEquation5thPowers.html
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– Gerry Myerson
Dec 31 '18 at 2:58
1
$begingroup$
If you have access to MathSciNet, you can search for papers that have included Guy's book as citations. This may help with your search.
$endgroup$
– JavaMan
Dec 31 '18 at 3:48
|
show 4 more comments
$begingroup$
I am preparing a paper on a^5 + b^5 = c^5 + d^5, and am having considerable difficulty in the literature review.
I don't see anything on StackExchange.
According to Wolfram's Mathworld, a book (Guy, 1994, page 140) includes the assertion that this equation has been checked to 10^26 with no solution.
(http://mathworld.wolfram.com/DiophantineEquation5thPowers.html) However, I don't have a copy of that book and it isn't free online.
I was a little surprised that 1994 is the most current reference on this problem that has been studied since at least 300 - 400 years ago.
Q: Is there any information available on this problem?
diophantine-equations
asked Dec 31 '18 at 1:08
L1-AL1-A
365
$endgroup$
I am preparing a paper on a^5 + b^5 = c^5 + d^5, and am having considerable difficulty in the literature review.
I don't see anything on StackExchange.
According to Wolfram's Mathworld, a book (Guy, 1994, page 140) includes the assertion that this equation has been checked to 10^26 with no solution.
(http://mathworld.wolfram.com/DiophantineEquation5thPowers.html) However, I don't have a copy of that book and it isn't free online.
I was a little surprised that 1994 is the most current reference on this problem that has been studied since at least 300 - 400 years ago.
Q: Is there any information available on this problem?
diophantine-equations
diophantine-equations
asked Dec 31 '18 at 1:08
L1-AL1-A
365
asked Dec 31 '18 at 1:08
L1-AL1-A
365
edited Jan 9 at 6:23
L1-A
asked Dec 31 '18 at 1:08
L1-AL1-A
365
asked Dec 31 '18 at 1:08
L1-AL1-A
365
asked Dec 31 '18 at 1:08
L1-AL1-A
365
365
closed as too broad by Eevee Trainer, Lord Shark the Unknown, Holo, José Carlos Santos, egreg Dec 31 '18 at 14:05
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as too broad by Eevee Trainer, Lord Shark the Unknown, Holo, José Carlos Santos, egreg Dec 31 '18 at 14:05
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/…
$endgroup$
– Will Jagy
Dec 31 '18 at 1:19
$begingroup$
I see. Table on page 453, it says for problem abbreviated 5.2.2 there are no known solutions, checked up to 2.0 * 10^14. As that paper was 1967, there could be substantially bigger searches in the intervening 50 years. It appears that your best bet is Andrew Bremner (1981) in the Journal of Number Theory, but it is clear he found no 5.2.2 or Guy and yur site would have said so.
$endgroup$
– Will Jagy
Dec 31 '18 at 1:28
1
$begingroup$
the book by Guy: books.google.com/… THE BAD NEWS is that it does not show the pages you want. Of course, those pages just report the stuff in those articles
$endgroup$
– Will Jagy
Dec 31 '18 at 2:02
1
$begingroup$
Mathworld cites the same result Guy cites, which I take as evidence that no one has checked further. mathworld.wolfram.com/DiophantineEquation5thPowers.html
$endgroup$
– Gerry Myerson
Dec 31 '18 at 2:58
1
$begingroup$
If you have access to MathSciNet, you can search for papers that have included Guy's book as citations. This may help with your search.
$endgroup$
– JavaMan
Dec 31 '18 at 3:48
|
show 4 more comments
$begingroup$
ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/…
$endgroup$
– Will Jagy
Dec 31 '18 at 1:19
$begingroup$
I see. Table on page 453, it says for problem abbreviated 5.2.2 there are no known solutions, checked up to 2.0 * 10^14. As that paper was 1967, there could be substantially bigger searches in the intervening 50 years. It appears that your best bet is Andrew Bremner (1981) in the Journal of Number Theory, but it is clear he found no 5.2.2 or Guy and yur site would have said so.
$endgroup$
– Will Jagy
Dec 31 '18 at 1:28
1
$begingroup$
the book by Guy: books.google.com/… THE BAD NEWS is that it does not show the pages you want. Of course, those pages just report the stuff in those articles
$endgroup$
– Will Jagy
Dec 31 '18 at 2:02
1
$begingroup$
Mathworld cites the same result Guy cites, which I take as evidence that no one has checked further. mathworld.wolfram.com/DiophantineEquation5thPowers.html
$endgroup$
– Gerry Myerson
Dec 31 '18 at 2:58
1
$begingroup$
If you have access to MathSciNet, you can search for papers that have included Guy's book as citations. This may help with your search.
$endgroup$
– JavaMan
Dec 31 '18 at 3:48
$begingroup$
ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/…
$endgroup$
– Will Jagy
Dec 31 '18 at 1:19
$begingroup$
ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/…
$endgroup$
– Will Jagy
Dec 31 '18 at 1:19
$begingroup$
I see. Table on page 453, it says for problem abbreviated 5.2.2 there are no known solutions, checked up to 2.0 * 10^14. As that paper was 1967, there could be substantially bigger searches in the intervening 50 years. It appears that your best bet is Andrew Bremner (1981) in the Journal of Number Theory, but it is clear he found no 5.2.2 or Guy and yur site would have said so.
$endgroup$
– Will Jagy
Dec 31 '18 at 1:28
$begingroup$
I see. Table on page 453, it says for problem abbreviated 5.2.2 there are no known solutions, checked up to 2.0 * 10^14. As that paper was 1967, there could be substantially bigger searches in the intervening 50 years. It appears that your best bet is Andrew Bremner (1981) in the Journal of Number Theory, but it is clear he found no 5.2.2 or Guy and yur site would have said so.
$endgroup$
– Will Jagy
Dec 31 '18 at 1:28
1
1
$begingroup$
the book by Guy: books.google.com/… THE BAD NEWS is that it does not show the pages you want. Of course, those pages just report the stuff in those articles
$endgroup$
– Will Jagy
Dec 31 '18 at 2:02
$begingroup$
the book by Guy: books.google.com/… THE BAD NEWS is that it does not show the pages you want. Of course, those pages just report the stuff in those articles
$endgroup$
– Will Jagy
Dec 31 '18 at 2:02
1
1
$begingroup$
Mathworld cites the same result Guy cites, which I take as evidence that no one has checked further. mathworld.wolfram.com/DiophantineEquation5thPowers.html
$endgroup$
– Gerry Myerson
Dec 31 '18 at 2:58
$begingroup$
Mathworld cites the same result Guy cites, which I take as evidence that no one has checked further. mathworld.wolfram.com/DiophantineEquation5thPowers.html
$endgroup$
– Gerry Myerson
Dec 31 '18 at 2:58
1
1
$begingroup$
If you have access to MathSciNet, you can search for papers that have included Guy's book as citations. This may help with your search.
$endgroup$
– JavaMan
Dec 31 '18 at 3:48
$begingroup$
If you have access to MathSciNet, you can search for papers that have included Guy's book as citations. This may help with your search.
$endgroup$
– JavaMan
Dec 31 '18 at 3:48
|
show 4 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Guy's book (Unsolved Problems in Number Theory, Second Ed.) was discussing the generalisation with equal sums of $m$ $s$th powers. The relevant paragraph (Chapter D, Section 1, page 140) reads:
Parametric solutions are known for equal sums of equal numbers of like powers,
$$sum_{i=1}^ma_i^s=sum_{i=1}^mb_i^s$$
with $a_i>0$, $b_1>0$, for $2leq sleq4$ and $m=2$ and for $s=5,6$ and $m=3$. Can a solution be found for $s=7$ and $m=4$? For $s=5,m=2$, it is not known if there is any nontrivial solution of $a^5+b^5=c^5+d^5$. Dick Lehmer once thought that there might be a solution with a sum of about $25$ decimal digits, but a search by Blair Kelly III yielded no nontrivial solution with sum $leq 1.02times10^{26}$.
The book does not cite Lehmer, but Kelly's paper is cited as: John B. Kelly, Two equal sums of three squares with equal products, Amer. Math. Monthly, 98(1991) 527-529; MR 92j:11025.
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
$endgroup$
$begingroup$
To the person who suggested that this was the wrong Kelly, I believe you are mistaken. The Kelly you gave works for the National Security Agency, USA, while this John B. Kelly is a professor at Arizona State University. You can find the paper here: jstor.org/stable/2324873
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– YiFan
Mar 31 at 22:08
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Guy's book (Unsolved Problems in Number Theory, Second Ed.) was discussing the generalisation with equal sums of $m$ $s$th powers. The relevant paragraph (Chapter D, Section 1, page 140) reads:
Parametric solutions are known for equal sums of equal numbers of like powers,
$$sum_{i=1}^ma_i^s=sum_{i=1}^mb_i^s$$
with $a_i>0$, $b_1>0$, for $2leq sleq4$ and $m=2$ and for $s=5,6$ and $m=3$. Can a solution be found for $s=7$ and $m=4$? For $s=5,m=2$, it is not known if there is any nontrivial solution of $a^5+b^5=c^5+d^5$. Dick Lehmer once thought that there might be a solution with a sum of about $25$ decimal digits, but a search by Blair Kelly III yielded no nontrivial solution with sum $leq 1.02times10^{26}$.
The book does not cite Lehmer, but Kelly's paper is cited as: John B. Kelly, Two equal sums of three squares with equal products, Amer. Math. Monthly, 98(1991) 527-529; MR 92j:11025.
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
$endgroup$
$begingroup$
To the person who suggested that this was the wrong Kelly, I believe you are mistaken. The Kelly you gave works for the National Security Agency, USA, while this John B. Kelly is a professor at Arizona State University. You can find the paper here: jstor.org/stable/2324873
$endgroup$
– YiFan
Mar 31 at 22:08
add a comment |
$begingroup$
Guy's book (Unsolved Problems in Number Theory, Second Ed.) was discussing the generalisation with equal sums of $m$ $s$th powers. The relevant paragraph (Chapter D, Section 1, page 140) reads:
Parametric solutions are known for equal sums of equal numbers of like powers,
$$sum_{i=1}^ma_i^s=sum_{i=1}^mb_i^s$$
with $a_i>0$, $b_1>0$, for $2leq sleq4$ and $m=2$ and for $s=5,6$ and $m=3$. Can a solution be found for $s=7$ and $m=4$? For $s=5,m=2$, it is not known if there is any nontrivial solution of $a^5+b^5=c^5+d^5$. Dick Lehmer once thought that there might be a solution with a sum of about $25$ decimal digits, but a search by Blair Kelly III yielded no nontrivial solution with sum $leq 1.02times10^{26}$.
The book does not cite Lehmer, but Kelly's paper is cited as: John B. Kelly, Two equal sums of three squares with equal products, Amer. Math. Monthly, 98(1991) 527-529; MR 92j:11025.
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
$endgroup$
$begingroup$
To the person who suggested that this was the wrong Kelly, I believe you are mistaken. The Kelly you gave works for the National Security Agency, USA, while this John B. Kelly is a professor at Arizona State University. You can find the paper here: jstor.org/stable/2324873
$endgroup$
– YiFan
Mar 31 at 22:08
add a comment |
$begingroup$
Guy's book (Unsolved Problems in Number Theory, Second Ed.) was discussing the generalisation with equal sums of $m$ $s$th powers. The relevant paragraph (Chapter D, Section 1, page 140) reads:
Parametric solutions are known for equal sums of equal numbers of like powers,
$$sum_{i=1}^ma_i^s=sum_{i=1}^mb_i^s$$
with $a_i>0$, $b_1>0$, for $2leq sleq4$ and $m=2$ and for $s=5,6$ and $m=3$. Can a solution be found for $s=7$ and $m=4$? For $s=5,m=2$, it is not known if there is any nontrivial solution of $a^5+b^5=c^5+d^5$. Dick Lehmer once thought that there might be a solution with a sum of about $25$ decimal digits, but a search by Blair Kelly III yielded no nontrivial solution with sum $leq 1.02times10^{26}$.
The book does not cite Lehmer, but Kelly's paper is cited as: John B. Kelly, Two equal sums of three squares with equal products, Amer. Math. Monthly, 98(1991) 527-529; MR 92j:11025.
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
$endgroup$
Guy's book (Unsolved Problems in Number Theory, Second Ed.) was discussing the generalisation with equal sums of $m$ $s$th powers. The relevant paragraph (Chapter D, Section 1, page 140) reads:
Parametric solutions are known for equal sums of equal numbers of like powers,
$$sum_{i=1}^ma_i^s=sum_{i=1}^mb_i^s$$
with $a_i>0$, $b_1>0$, for $2leq sleq4$ and $m=2$ and for $s=5,6$ and $m=3$. Can a solution be found for $s=7$ and $m=4$? For $s=5,m=2$, it is not known if there is any nontrivial solution of $a^5+b^5=c^5+d^5$. Dick Lehmer once thought that there might be a solution with a sum of about $25$ decimal digits, but a search by Blair Kelly III yielded no nontrivial solution with sum $leq 1.02times10^{26}$.
The book does not cite Lehmer, but Kelly's paper is cited as: John B. Kelly, Two equal sums of three squares with equal products, Amer. Math. Monthly, 98(1991) 527-529; MR 92j:11025.
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
answered Dec 31 '18 at 2:19
YiFanYiFan
5,6702831
5,6702831
$begingroup$
To the person who suggested that this was the wrong Kelly, I believe you are mistaken. The Kelly you gave works for the National Security Agency, USA, while this John B. Kelly is a professor at Arizona State University. You can find the paper here: jstor.org/stable/2324873
$endgroup$
– YiFan
Mar 31 at 22:08
add a comment |
$begingroup$
To the person who suggested that this was the wrong Kelly, I believe you are mistaken. The Kelly you gave works for the National Security Agency, USA, while this John B. Kelly is a professor at Arizona State University. You can find the paper here: jstor.org/stable/2324873
$endgroup$
– YiFan
Mar 31 at 22:08
$begingroup$
To the person who suggested that this was the wrong Kelly, I believe you are mistaken. The Kelly you gave works for the National Security Agency, USA, while this John B. Kelly is a professor at Arizona State University. You can find the paper here: jstor.org/stable/2324873
$endgroup$
– YiFan
Mar 31 at 22:08
$begingroup$
To the person who suggested that this was the wrong Kelly, I believe you are mistaken. The Kelly you gave works for the National Security Agency, USA, while this John B. Kelly is a professor at Arizona State University. You can find the paper here: jstor.org/stable/2324873
$endgroup$
– YiFan
Mar 31 at 22:08
add a comment |
$begingroup$
ams.org/journals/mcom/1967-21-099/S0025-5718-1967-0222008-0/…
$endgroup$
– Will Jagy
Dec 31 '18 at 1:19
$begingroup$
I see. Table on page 453, it says for problem abbreviated 5.2.2 there are no known solutions, checked up to 2.0 * 10^14. As that paper was 1967, there could be substantially bigger searches in the intervening 50 years. It appears that your best bet is Andrew Bremner (1981) in the Journal of Number Theory, but it is clear he found no 5.2.2 or Guy and yur site would have said so.
$endgroup$
– Will Jagy
Dec 31 '18 at 1:28
1
$begingroup$
the book by Guy: books.google.com/… THE BAD NEWS is that it does not show the pages you want. Of course, those pages just report the stuff in those articles
$endgroup$
– Will Jagy
Dec 31 '18 at 2:02
1
$begingroup$
Mathworld cites the same result Guy cites, which I take as evidence that no one has checked further. mathworld.wolfram.com/DiophantineEquation5thPowers.html
$endgroup$
– Gerry Myerson
Dec 31 '18 at 2:58
1
$begingroup$
If you have access to MathSciNet, you can search for papers that have included Guy's book as citations. This may help with your search.
$endgroup$
– JavaMan
Dec 31 '18 at 3:48