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There is an easier example in

where Kap disposed of the concern with the brief "(it is easy to see that the assumption of no congruence obstructions is satisfied)."

The example is, given a positive prime $p \equiv 1 \pmod 4,$ there is no solution in integers $x,y,z$ to $$x^2 + y^2 + z^9 = 216 p^3$$

Robert C. Vaughan wrote to Kap (prior to publication) in appreciation, there was something involved that "could not be detected p-adically." I forget what, it has been years. But we did well, Vaughan got an early draft in time to include the example in the second edition of
The Hardy-Littlewood Method.

Later for some reason I looked at negative targets, with the same primes I believe it turned out that there were no integer solutions to $$x^2 + y^2 + z^9 = -8 p^3.$$

The significance of the example is not so much as a single Diophantine equation, rather as a Diophantine representation problem in the general vicinity of the Waring problem, but with mixed exponents: given nonegative integer variables $x,y,z$ and exponents $a,b,c \geq 2,$ and given the polynomial $f(x,y,z) =x^a + y^b + z^c,$ if $f(x,y,z)$ represents every positive integer $p$-adically and if $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} > 1,$$ does $f(x,y,z)$ integrally represent all sufficiently large integers? The answer is no for the problem as stated, but the counterexamples depend heavily on factorization, and in the end upon composition of binary forms. As this is also the mechanism underlying the simplest examples of spinor exceptional integers for positive ternary quadratic forms, it is natural to ask whether there is some relatively easy formalism that adds "factorization obstructions" to the well-studied "congruence obstructions."

See:

http://zakuski.math.utsa.edu/~jagy/Vaughan.pdf

http://en.wikipedia.org/wiki/Waring's_problem

1

There is an easier example in

where Kap disposed of the concern with the brief "(it is easy to see that the assumption of no congruence obstructions is satisfied)."

The example is, given a positive prime $p \equiv 1 \pmod 4,$ there is no solution in integers $x,y,z$ to $$x^2 + y^2 + z^9 = 216 p^3$$

Robert C. Vaughan wrote to Kap (prior to publication) in appreciation, there was something involved that "could not be detected p-adically." I forget what, it has been years. But we did well, Vaughan got an early draft in time to include the example in the second edition of
The Hardy-Littlewood Method.

Later for some reason I looked at negative targets, with the same primes I believe it turned out that there were no integer solutions to $$x^2 + y^2 + z^9 = -8 p^3.$$