Diophantine equation 3^a+1=3^b+5^c Planned maintenance scheduled April 23, 2019 at 23:30 UTC...
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
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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 34 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 34 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 34 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 34 mins ago
kawakawa
1707
asked 34 mins ago
kawakawa
1707
asked 34 mins ago
kawakawa
1707
asked 34 mins ago
kawakawa
1707
asked 34 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.
answered 13 mins ago
LuciaLucia
34.9k5151177
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
9 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
$begingroup$
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.
answered 13 mins ago
LuciaLucia
34.9k5151177
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
9 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.
answered 13 mins ago
LuciaLucia
34.9k5151177
$endgroup$
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
9 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.
answered 13 mins ago
LuciaLucia
34.9k5151177
$endgroup$
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.
answered 13 mins ago
LuciaLucia
34.9k5151177
answered 13 mins ago
LuciaLucia
34.9k5151177
answered 13 mins ago
LuciaLucia
34.9k5151177
answered 13 mins ago
LuciaLucia
34.9k5151177
34.9k5151177
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
9 mins ago
add a comment |
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
9 mins ago
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
9 mins ago
$begingroup$
Lucia, many thanks for the paper.
$endgroup$
– kawa
9 mins ago
add a comment |
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