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What are the positive integer solutions to the equation

$$2^a + 3^b = 5^c$$

Of course $(1,\space 1, \space 1)$ is a solution.

  • 0
    Anything else would violate [Beal's conjecture](http://en.wikipedia.org/wiki/Beal%27s_conjecture) so the solutions given are likely it.2012-11-01
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    This result should solve the case that $a$ is even: "It is also true that the equation $3^x+4^y=5^z$ has no solutions in natural numbers except $x = y = z = 2$." Sierpinski, Elementary Number Theory, [p.40](http://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA40). Refernces given there: W. Sierpinski: O rownaniu $3^x+4^y=5^z$, {Wiadom. Mat.}, 1 (1956), 194-195. Nagell T.: Sur une classe d'equations exponentielles, Ark. Mat. 3 (1958) 569-582.2012-11-02

4 Answers 4

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If $a=0$, then it is clear there are no solutions. If $b=0$, then we need $2^a + 1 = 5^c$. It is easy to show in this case $a=2,c=1$ is the only solution by showing that we need $2^{a-2}|c$. When $c=0$ there are obviously no solutions.

Suppose $a=1$. Then $2 + 3^b = 5^c$ only has the solution of $b=1,c=1$. To show this check modulo $275$ to deduce $c=1$ and thus $b=1$ is forced.

Now, suppose $a \ge 3$. Then remark that by checking modulo $4$ we need $b$ to be even so let $b = 2b'$. So let's solve $2^a + 3^{2b'} = 5^c$. Checking modulo $8$ we get $c$ is even so let $c = 2c'$. Then: $$2^a = (5^{c'} - 3^{b'})(5^{c'}+3^{b'})$$ We get $5^{c'} - 3^{b'} = 2^m, 5^{c'}+3^{b'} = 2^n$ for some $m,n$. But then $2 \cdot 5^{c'} = 2^m + 2^n$, forcing $m=1$. Thus we need $5^{c'} - 3^{b'} = 2$. But we already showed this only has the solution $b' = 1, c' = 1$. Thus it follows the only solution where $a \ge 2$ is with $a=4, b = 2, c= 2$.

Putting every together, we have proven the only solutions are $(2,0,1), (1,1,1), (4,2,2)$

EDIT: I realize I forgot to do the case of $a=2$. So we need to solve $4 + 3^{2b'} = 5^c$. Modulo $275$ happens to work again to force $c=1$ and thus we get no solutions when $b$ is nonzero.

  • 0
    What is the purpose of using modulo 275? Also, you should check out Størmer's theorem and Baker's theorem. They can simplify the above proof to maybe 2 paragraphs at most, and neither of them require modular arithmetic.2012-12-13
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    The purpose of modulo $275$ is to deduce $c=1$ (note that $25|275$). Also considering this problem is tagged as Elementary Number Theory, I don't think the question poser was looking for solutions using that kind of heavy machinery.2012-12-13
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    Would it have worked if you used 175 instead? I'm still not understanding if 275 is an arbitrary choice or not.2012-12-14
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    275 is decently arbitrary, the reason why it works is because the order of 3 and 5 modulo 11 is fairly low since they are both quadratic residues so the chance of it working is much higher than other primes such as 7.2012-12-14
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    As a side note, mod 252 is easier to calculate by hand than mod 275. The powers of 3 mod 252 are 9, 27, 81, 243, 225, 171, and the powers of 5 mod 252 are 1, 5, 25, 125, 121, 101.2018-08-24
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Another solution is $2^4 + 3^2 = 5^2$. That's probably all.

2

case : $b>2$ and $c>2$

A calculus with computer, modulo $N=2372625=5^3\times 19 \times 37 \times 3^3$, in $H=\mathbb Z/N\mathbb Z $ give

$\text{card}(<2>)=900$, $\text{card}(3^3\times <3>)=900$, $\text{card}(5^3\times<5>)=36$.

And $(<2>+(3^3\times <3>)) \cap 5^3\times <5>=\emptyset$

with $1\in $ the subset of $H$ generated by $g$ with times ($\times$)

So if $b>2$ and $c>2$, there are not solutions.

case : $b\leq 2$ and $c>2$

Here we choose $N=5^3\times (2^{25}-1)$

so $(<2>+\{1,3,9\}) \cap 5^3\times <5>=\emptyset$

So if $b\leq 2$ and $c>2$ no solution.

case : $c \leq 2$ three solutions

Then there are only 3 solutions, not any more.

  • 0
    Where did $2372625$ come from??2017-07-23
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    I have choose (n=19*37) for i,j,k is small for 2^i mod n=3^j mod n=5^k mod n=1, n|gcd(2^i-1,3^j-1,5^k-1) is big and i,j,k is small, To minimize the risk of collision @AkivaWeinberger2017-07-23
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Here is a little program in Python:

for k in range(10): for l in range(10):     for m in range(10):         if 2**k + 3**l == 5**m:             print (k, l, m) 

Its output is as follows

    python check.py 1 1 1 2 0 1 4 2 2 

This yields these three equations.

$$2 + 3 = 5$$ $$2^2 + 3^0 = 5^1$$ $$2^4 + 3^2 = 5^2.$$

And that's all folks! Swap 100 for 10 and the result is the same. Cooking up an analytical proof of we have done is not hard.

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    the second solution contains a $0$2012-11-01
  • 0
    C'est vrai. Ignore it. I just was checking the corner cases.2012-11-01
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    I just checked that there are no other solutions for $a, b < 600$.2012-11-01
  • 2
    Could there be solutions > 600?2012-11-01