1979 USAMO Problems/Problem 1

Problem

Determine all non-negative integral solutions $(n_1,n_2,\dots , n_{14})$ if any, apart from permutations, of the Diophantine Equation $n_1^4+n_2^4+\cdots +n_{14}^4=1599$ .

Solution 1

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

Recall that $n_i^4\equiv 0,1\bmod{16}$ for all integers $n_i$ . Thus the sum we have is anything from 0 to 14 modulo 16. But $1599\equiv 15\bmod{16}$ , and thus there are no integral solutions to the given Diophantine equation.

Solution 2

By AM-GM, $\dfrac{n^{4}_1+...+n^{4}_{14}}{14}\geq(n_1...n_{14})^{2/7}$ . Assume there exist nonnegative integers $n_i$ which satisfy the given equation. Then $(n_1...n_{14})\leq(\dfrac{1599}{14}^{7/2}\)<2$ (Error compiling LaTeX. Unknown error_msg). So $n_1...n_{14}=0$ or $1$ , clearly a contradiction. Thus there are no solutions to the given equation.

1979 USAMO (Problems • Resources)
Preceded by First Question	Followed by Problem 2
1 • 2 • 3 • 4 • 5
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1979 USAMO Problems/Problem 1

Contents

Problem

Solution 1

Solution 2

See also