## View abstract

### Session S09 - Number Theory in the Americas

Wednesday, July 14, 13:00 ~ 13:30 UTC-3

## On some exponential Diophantine equations involving sequences

### Alain Togbe

#### Purdue U. Northwest, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak866f3b10d195dd032b9eff5a3e8424db').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy866f3b10d195dd032b9eff5a3e8424db = '&#97;t&#111;gb&#101;' + '&#64;'; addy866f3b10d195dd032b9eff5a3e8424db = addy866f3b10d195dd032b9eff5a3e8424db + 'pnw' + '&#46;' + '&#101;d&#117;'; var addy_text866f3b10d195dd032b9eff5a3e8424db = '&#97;t&#111;gb&#101;' + '&#64;' + 'pnw' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak866f3b10d195dd032b9eff5a3e8424db').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy866f3b10d195dd032b9eff5a3e8424db + '\'>'+addy_text866f3b10d195dd032b9eff5a3e8424db+'<\/a>';

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0=0,~F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for all $n\geq 2$. We consider the Diophantine equation $$F_n^x+F_{n+1}^x+\cdots+F_{n+k-1}^x=F_m$$ and determine all nonnegative integer solutions $(x, m, n, k)$ of this equation. Moreover, if we replace the Fibonacci sequence by the Pell sequence and the Padovan sequence, we solve the corresponding equations.

Joint work with E. Tchammou (IMSP, Benin) and F. Luca (Wits, South Africa and UNAM, Mexico).