Diophantine equation $3^a+1=3^b+5^c$
$begingroup$
This is not a research problem, but challenging enough that I've decided to post it in here:
Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$
nt.number-theory diophantine-equations elementary-proofs
edited Apr 22 at 5:31
TheSimpliFire
12510
asked Apr 22 at 0:23
kawakawa
2188
$endgroup$
add a comment |
$begingroup$
This is not a research problem, but challenging enough that I've decided to post it in here:
Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$
nt.number-theory diophantine-equations elementary-proofs
edited Apr 22 at 5:31
TheSimpliFire
12510
asked Apr 22 at 0:23
kawakawa
2188
$endgroup$
3
$begingroup$
Papers are still published on such questions (Lucia gave two examples, and there are others), so it's research-level (and thus fair game for Mathoverflow) even if it did not arise in your own research.
$endgroup$
– Noam D. Elkies
Apr 22 at 2:47
2
$begingroup$
But if it's a puzzle to which OP already knows the answer, then I'd say it's not appropriate for MO.
$endgroup$
– Gerry Myerson
Apr 22 at 12:35
$begingroup$
NoamD.Elkies: thanks for your comment. GerryMyerson: thanks. Had I known that these type of Diophantine equations are still an active area of research, would have likely phrased the problem that way. Still trying to digest the concept --- as there is a section (tag) with elementary proofs, that are contest problems, as opposed to research problems, but challenging enough that people still post.
$endgroup$
– kawa
Apr 22 at 14:14
add a comment |
$begingroup$
This is not a research problem, but challenging enough that I've decided to post it in here:
Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$
nt.number-theory diophantine-equations elementary-proofs
edited Apr 22 at 5:31
TheSimpliFire
12510
asked Apr 22 at 0:23
kawakawa
2188
$endgroup$
This is not a research problem, but challenging enough that I've decided to post it in here:
Determine all triples $(a,b,c)$ of non-negative integers, satisfying
$$
1+3^a = 3^b+5^c.
$$
nt.number-theory diophantine-equations elementary-proofs
nt.number-theory diophantine-equations elementary-proofs
edited Apr 22 at 5:31
TheSimpliFire
12510
asked Apr 22 at 0:23
kawakawa
2188
edited Apr 22 at 5:31
TheSimpliFire
12510
asked Apr 22 at 0:23
kawakawa
2188
edited Apr 22 at 5:31
TheSimpliFire
12510
edited Apr 22 at 5:31
TheSimpliFire
12510
edited Apr 22 at 5:31
TheSimpliFire
12510
12510
asked Apr 22 at 0:23
kawakawa
2188
asked Apr 22 at 0:23
kawakawa
2188
asked Apr 22 at 0:23
kawakawa
2188
2188
3
$begingroup$
Papers are still published on such questions (Lucia gave two examples, and there are others), so it's research-level (and thus fair game for Mathoverflow) even if it did not arise in your own research.
$endgroup$
– Noam D. Elkies
Apr 22 at 2:47
2
$begingroup$
But if it's a puzzle to which OP already knows the answer, then I'd say it's not appropriate for MO.
$endgroup$
– Gerry Myerson
Apr 22 at 12:35
$begingroup$
NoamD.Elkies: thanks for your comment. GerryMyerson: thanks. Had I known that these type of Diophantine equations are still an active area of research, would have likely phrased the problem that way. Still trying to digest the concept --- as there is a section (tag) with elementary proofs, that are contest problems, as opposed to research problems, but challenging enough that people still post.
$endgroup$
– kawa
Apr 22 at 14:14
add a comment |
3
$begingroup$
Papers are still published on such questions (Lucia gave two examples, and there are others), so it's research-level (and thus fair game for Mathoverflow) even if it did not arise in your own research.
$endgroup$
– Noam D. Elkies
Apr 22 at 2:47
2
$begingroup$
But if it's a puzzle to which OP already knows the answer, then I'd say it's not appropriate for MO.
$endgroup$
– Gerry Myerson
Apr 22 at 12:35
$begingroup$
NoamD.Elkies: thanks for your comment. GerryMyerson: thanks. Had I known that these type of Diophantine equations are still an active area of research, would have likely phrased the problem that way. Still trying to digest the concept --- as there is a section (tag) with elementary proofs, that are contest problems, as opposed to research problems, but challenging enough that people still post.
$endgroup$
– kawa
Apr 22 at 14:14
3
3
$begingroup$
Papers are still published on such questions (Lucia gave two examples, and there are others), so it's research-level (and thus fair game for Mathoverflow) even if it did not arise in your own research.
$endgroup$
– Noam D. Elkies
Apr 22 at 2:47
$begingroup$
Papers are still published on such questions (Lucia gave two examples, and there are others), so it's research-level (and thus fair game for Mathoverflow) even if it did not arise in your own research.
$endgroup$
– Noam D. Elkies
Apr 22 at 2:47
2
2
$begingroup$
But if it's a puzzle to which OP already knows the answer, then I'd say it's not appropriate for MO.
$endgroup$
– Gerry Myerson
Apr 22 at 12:35
$begingroup$
But if it's a puzzle to which OP already knows the answer, then I'd say it's not appropriate for MO.
$endgroup$
– Gerry Myerson
Apr 22 at 12:35
$begingroup$
NoamD.Elkies: thanks for your comment. GerryMyerson: thanks. Had I known that these type of Diophantine equations are still an active area of research, would have likely phrased the problem that way. Still trying to digest the concept --- as there is a section (tag) with elementary proofs, that are contest problems, as opposed to research problems, but challenging enough that people still post.
$endgroup$
– kawa
Apr 22 at 14:14
$begingroup$
NoamD.Elkies: thanks for your comment. GerryMyerson: thanks. Had I known that these type of Diophantine equations are still an active area of research, would have likely phrased the problem that way. Still trying to digest the concept --- as there is a section (tag) with elementary proofs, that are contest problems, as opposed to research problems, but challenging enough that people still post.
$endgroup$
– kawa
Apr 22 at 14:14
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
$$
ap^x + bq^y = c+ dp^z q^w
$$
has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.
Following a reference from Skinner's paper, Theorem 4.01 of Brenner and Foster gives an explicit treatment of the equation
$$
3^a + 7^b=3^c+5^d,
$$
which completely resolves the problem in this question (take $b=0$). Their proof is elementary, and the only non-trivial solution to the equation in the question is $3^3+1 = 3 + 5^2$.
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
Apr 22 at 0:48
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
$$
ap^x + bq^y = c+ dp^z q^w
$$
has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.
Following a reference from Skinner's paper, Theorem 4.01 of Brenner and Foster gives an explicit treatment of the equation
$$
3^a + 7^b=3^c+5^d,
$$
which completely resolves the problem in this question (take $b=0$). Their proof is elementary, and the only non-trivial solution to the equation in the question is $3^3+1 = 3 + 5^2$.
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
Apr 22 at 0:48
add a comment |
$begingroup$
I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
$$
ap^x + bq^y = c+ dp^z q^w
$$
has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.
Following a reference from Skinner's paper, Theorem 4.01 of Brenner and Foster gives an explicit treatment of the equation
$$
3^a + 7^b=3^c+5^d,
$$
which completely resolves the problem in this question (take $b=0$). Their proof is elementary, and the only non-trivial solution to the equation in the question is $3^3+1 = 3 + 5^2$.
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
Apr 22 at 0:48
add a comment |
$begingroup$
I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
$$
ap^x + bq^y = c+ dp^z q^w
$$
has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.
Following a reference from Skinner's paper, Theorem 4.01 of Brenner and Foster gives an explicit treatment of the equation
$$
3^a + 7^b=3^c+5^d,
$$
which completely resolves the problem in this question (take $b=0$). Their proof is elementary, and the only non-trivial solution to the equation in the question is $3^3+1 = 3 + 5^2$.
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
$endgroup$
I can't resist this: The young Chris Skinner showed that if $a$, $b$, $c$, $d$ are fixed positive integers, and $p$ and $q$ are positive coprime integers then the equation
$$
ap^x + bq^y = c+ dp^z q^w
$$
has a bounded number of solutions in $(x,y,z,w)$ and that a bound on these could be computed (and the equation solved in practice). This solves (in principle) the more general equation $1+3^a 5^d = 3^b+ 5^c$. Anyway, there is a large literature around such exponential diophantine equations, and Skinner's paper will give some references.
Following a reference from Skinner's paper, Theorem 4.01 of Brenner and Foster gives an explicit treatment of the equation
$$
3^a + 7^b=3^c+5^d,
$$
which completely resolves the problem in this question (take $b=0$). Their proof is elementary, and the only non-trivial solution to the equation in the question is $3^3+1 = 3 + 5^2$.
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
edited Apr 22 at 1:13
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
answered Apr 22 at 0:44
LuciaLucia
35.2k5152179
35.2k5152179
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
Apr 22 at 0:48
add a comment |
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
Apr 22 at 0:48
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
Apr 22 at 0:48
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
Apr 22 at 0:48
add a comment |
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$begingroup$
Papers are still published on such questions (Lucia gave two examples, and there are others), so it's research-level (and thus fair game for Mathoverflow) even if it did not arise in your own research.
$endgroup$
– Noam D. Elkies
Apr 22 at 2:47
2
$begingroup$
But if it's a puzzle to which OP already knows the answer, then I'd say it's not appropriate for MO.
$endgroup$
– Gerry Myerson
Apr 22 at 12:35
$begingroup$
NoamD.Elkies: thanks for your comment. GerryMyerson: thanks. Had I known that these type of Diophantine equations are still an active area of research, would have likely phrased the problem that way. Still trying to digest the concept --- as there is a section (tag) with elementary proofs, that are contest problems, as opposed to research problems, but challenging enough that people still post.
$endgroup$
– kawa
Apr 22 at 14:14