Tabăra MathTime - Problema 9, Ziua I - JUNIORI
- Laurențiu Ploscaru
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Tabăra MathTime - Problema 9, Ziua I - JUNIORI
Demonstrați că ecuația $x^4+y^4+z^4=2002^t$ are o infinitate de soluții în $\Bbb{N}$.
People are strange when you're a stranger,
Faces look ugly when you're alone.
Women seem wicked when you're unwanted,
Streets are uneven when you're down.
Faces look ugly when you're alone.
Women seem wicked when you're unwanted,
Streets are uneven when you're down.
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Re: Tabăra MathTime - Problema 9, Ziua I - JUNIORI
Se observa ca $2002=3^4+5^4+6^4$(*), deci o solutie a ecuatiei date este x=3;y=5;z=6 si t=1. Inmultind (*) cu $2002^{4k}$, k natural, obtinem :
$2002^{4k+1}=(3*2002^k)^4+(5*2002^k)^4+(6*2002^k)^4$,
de unde concluzia.
$2002^{4k+1}=(3*2002^k)^4+(5*2002^k)^4+(6*2002^k)^4$,
de unde concluzia.
Nimic nu-i niciodata asa de simplu cum pare.