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Diophantine equation 3^a+1=3^b+5^c
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
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Unicorn Meta Zoo #1: Why another podcast?Transforming a Diophantine equation to an elliptic curveNon-negative integer solutions of a single Linear Diophantine EquationDiophantine problemDoes the following Diophantine equation have nontrivial rational solutions?Help with this Diophantine equationHelp with this system of Diophantine equationsThe Theory of Transfinite Diophantine EquationsFind a distinct postive integer solution to this $xyzw=504(x^2+y^2+z^2+w^2)$ diophantine equationExponential diophantine equation systemCombination of $k$-powers and divisibility
$begingroup$
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
asked 59 mins ago
kawakawa
1707
$endgroup$
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
asked 59 mins ago
kawakawa
1707
$endgroup$
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
asked 59 mins ago
kawakawa
1707
$endgroup$
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
asked 59 mins ago
kawakawa
1707
asked 59 mins ago
kawakawa
1707
asked 59 mins ago
kawakawa
1707
asked 59 mins ago
kawakawa
1707
asked 59 mins ago
kawakawa
1707
1707
add a comment |
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 38 mins ago
LuciaLucia
35k5151177
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
34 mins ago
add a comment |
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1 Answer
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1 Answer
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active
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active
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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 38 mins ago
LuciaLucia
35k5151177
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
34 mins ago
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 38 mins ago
LuciaLucia
35k5151177
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
34 mins ago
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 38 mins ago
LuciaLucia
35k5151177
$endgroup$
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 38 mins ago
LuciaLucia
35k5151177
edited 9 mins ago
answered 38 mins ago
LuciaLucia
35k5151177
answered 38 mins ago
LuciaLucia
35k5151177
answered 38 mins ago
LuciaLucia
35k5151177
35k5151177
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
34 mins ago
add a comment |
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
34 mins ago
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
34 mins ago
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
34 mins ago
add a comment |
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