Solutions To The Diophantine #1

by AlastorMoody, Sep 4, 2019, 9:06 AM

Theorem: For $p,q >1 \in \mathbb{Z}$
$$3^p-2^q=1 ~ ~ ~ \qquad \qquad - - - \text{(1)}$$$$2^p-3^q=1 ~ ~ ~ \qquad \qquad - - - \text{(2)}$$Equation $\text{(1)}$ has solutions: $\boxed{(p,q) \equiv (2,3)}$ whereas Equation $\text{(2)}$ has no solutions
Proof (1): If $p=2k+1$, for $k \in \mathbb{N}$
$$2^q=3^p-1=3 \cdot 9^k-1 \equiv 2 \pmod{4}$$Which is not true for $ q >1$. If $p=2k$ for $k \in \mathbb{N}$ $\implies$ $2^q =3^p-1=(3^k-1)(3^k+1)$ $\implies$ $3^k-1=2^a$ and $3^k+1=2^b$ $\implies$ $2^b-2^a$ $=$ $(3^k+1)$ $-$ $(3^k-1)$ $=$ $2$ $\implies$ $\boxed{(p,q) \equiv (2,3)} \qquad \blacksquare$
Proof (2): Taking $\pmod{8}$ $\implies$ $3^q \equiv 7 \pmod{8} $ for $p \geq 3 $ $\implies$ No solution for $q$. Hence, $p=2$ which gives no solution for $q>1$ $\qquad \blacksquare$
Problems Involved
JBMO SL 2009 N1 wrote:
Solve in non-negative integers the equation: $ 2^a3^b+9 = c^2$
INMO 1992 P8 wrote:
Determine all pairs $(m,n)$ of positive integers for which $2^{m} + 3^{n}$ is a perfect square.
ISL 1991 P17 wrote:
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x + 4^y = 5^z.$
This post has been edited 5 times. Last edited by AlastorMoody, Oct 11, 2019, 7:15 PM

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3 Comments

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Hmmm....The first non-geo post in your blog. And it turns out to be NT. Am I proud of you? Hell yeah!

Edit (AlastorMoody): Oh Lord Hansugami :omighty: You're the only & only Legend :omighty:
This post has been edited 1 time. Last edited by AlastorMoody, Oct 14, 2019, 5:42 AM

by hansu, Oct 12, 2019, 2:42 PM

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You might wanna look up Catalan's Conjecture/Mihailescu's Theorem. ;)

Edit (AlastorMoody): Write Mihailescu's Theorem on RMO & lemme see if they give you marks lol :P
This post has been edited 1 time. Last edited by AlastorMoody, Oct 14, 2019, 5:40 AM

by MathBoy23, Oct 12, 2019, 3:36 PM

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No geo bash but still mod basher :mad:

U s0 pr0 :) Geo $\cup$ NT = $\infty_{\mathcal{P}}$

by o_i-SNAKE-i_o, Oct 26, 2019, 8:19 AM

I'll talk about all possible non-sense :D

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    by crazyeyemoody907, Oct 31, 2023, 1:55 AM

  • @below I wish I started earlier / didn't have to do JEE and leave oly way before I could study conics and projective stuff which I really wanted to study :( . Huh, life really sucks when u are forced due to peer pressure to read sh_t u dont want to read

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  • But even if he went to jee he could continue with this.

    Doing JEE(and completely leaving oly) seems like a insult to the oly math he knows

    by HoRI_DA_GRe8, Feb 11, 2022, 2:11 PM

  • Ohhh did he go for JEE? Good for him, bad for us :sadge:. Hmmm so that is the reason why he is inactive
    Btw @below finally everyone falls to the monopoly of JEE :) Coz IIT's are the best in India.

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