Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Question

Conjecture:

There does not exist a non-integer $x$ such that

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

where $a,b,c$ are all integers.

I'm aware that the similar question

There does not exist a non-integer $y$ such that

$$2^y=A$$ $$3^y=B$$

where $A,B$ are all integers.

is a famous unsolved problem.

(evidence in the corrolaries here or here)

My idea was that the addition of the condition $5^x$ made the problem easier and therefore solvable.

Why do you think this additional criterion makes it easier? Do you have anything concrete to add? — Ian Coley, Jan 01 '15 at 21:00
Its like a system of linear equations I guess. The more equations you have , the easier to solve , or show there is no solution. — mick, Jan 01 '15 at 21:02
AFAIK, the $2$, $3$, and $5$ version isn't an unsolved problem. There was a question on MSE about a week ago asking "if $n^x$ is an integer for all positive integers $n$, then $x$ is an integer". One of the comments suggested a link providing a proof that $2$, $3$, and $5$ are sufficient, hence providing a proof of this problem. I'll try to find that question later. — Edward Jiang, Jan 01 '15 at 21:03
See http://mathoverflow.net/questions/17560/if-2x-and-3x-are-integers-must-x-be-as-well — gtrrebel, Jan 01 '15 at 21:04

Will Jagy · Answer 1 · 2015-01-08T04:17:18.410

8

Theorem of Siegel that for real $\lambda$ and distinct primes $p,q,r,$ the numbers $$ p^\lambda, \; q^\lambda, \; r^\lambda, $$ cannot all be rational unless $\lambda$ is an integer.

See page 455 in Alaoglu and Erdos, On Highly Composite and Similar Numbers (1944), also chapter 2 of Introduction to Transcendental Numbers by S. Lang.

I see there is also a book Transcendental Numbers by our hero, C.L. Siegel, that could easily give the first polished discussion of the result, which is just what we call a "personal communication" in the Alaoglu Erdos article.

I do not have the Lang book from the library yet. Apparently this covers the same material, and is certainly from the same time: LANG PDF

From the book, now clear how it relates, see Corollary 1, with the definition of multiplicatively independent middle of page 8 :

enter image description here

This (Theorem 1, I guess) is this : http://en.wikipedia.org/wiki/Six_exponentials_theorem

edited Jan 08 '15 at 04:17

answered Jan 01 '15 at 22:38

Will Jagy

139,541

IS there no short answer ?? – mick Jan 06 '15 at 23:41
2

@mick, I do not see why there would be. I requested a copy of Lang from the library, not here yet. Most problems in number theory that are not constructed as exercises for books are extremely difficult, or currently impossible. It takes knowledge to make an approachable question. – Will Jagy Jan 06 '15 at 23:59
It's the Six Exponentials Theorem, and no short proof is known, nor is any expected. – Gerry Myerson Jan 08 '15 at 04:43
Compare https://www.sixflags.com/america
http://en.wikipedia.org/wiki/Six_Days_Seven_Nights

http://en.wikipedia.org/wiki/Six_degrees_of_separation
– Will Jagy Jan 08 '15 at 05:00

Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

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