There are some corollaries; perhaps these could be used to work backwards, building up a solution from smaller partials: d^3 - g^3 = e^3 - h^3 = f^3 - i^3, and the system d^3 - e^3 = g^3 - h^3, d^3 - f^3 = g^3 - i^3, which is a smaller version of your goal. ------------ A New Kind of Industrial Mathematics Consider a related subproblem: providing interesting sets of integers with prescribed density. We have primes with density falloff 1/logN, squares with density falloff N^-1/2, cubes with N^-2/3, powers of 2 with falloff logN/N, products of two primes with falloff loglogN/logN, etc. We can make combinations of these to create guesses like "every odd number is the sum of a prime and a power of 2" or "every even number is the sum of a prime and Mersenne-prime exponent". We can tune these to be truer by adding another thin set like 2^2^N, or noticing modular restrictions (primes are mostly odd), or saying N>1000, or "almost all N" (meaning exceptions are thin, -> 0%). We could introduce meta-sets, such as "numbers that are NOT the sum of three squares". With very little investment of fact, a whole bushel of conjecture! (borrowing from Mark Twain). I can identify one application of this sort of pseudo-theorem: Sometimes, as part of a larger algorithm, I'll need to say "There are lots of primes with properties A,B, & C; search until you find one." Or, I'll cheerfully assume the existence of a small primitive root. For engineering applications, this is accepted behavior, even though the rigor is absent. There's an obvious tendency to regard these assertions as mere annoyances, since we have no hope of proof for most of them. Any refutation will usually be based on trivial arguments, such as a numerical counterexample, or a simple mod 12 argument. Usually the refutations can be tuned away, frustrating the refuter. I'll argue for a more "industrial" approach: We can generate millions of these guesses mechanically, classify them mechanically, run the usual-suspect numerical tests, and mark the survivors as "provisionally true". We could generate some implications automatically, such as Goldbach-even2 -> Goldbach-odd3. The idea is to create a database somehwat like the sequence database or the real-number database. When confronted with a question, we could consult the database, hoping for a match or near-match that could cast light on our question. We could apply data-mining methods to look for generalizations that are actually interesting. This applies modern technology to the old idea of warehousing results that might be useful in the next century. Rich --------- Quoting Christian Boyer <cboyer@club-internet.fr>:

I am looking for at least one integer solution of this system: a^3 - d^3 = b^3 - e^3 = c^3 - f^3 a^3 - g^3 = b^3 - h^3 = c^3 - i^3

Quite easy to find near solutions, for example: 165^3 - 72^3 = 178^3 - 115^3 = 162^3 - 51^3 165^3 - 618^3 = 178^3 - 619^3 = 162^3 - 235788435 Unfortunately 235788435 is not a cube... It seems that there is no solution with integers < 300000^3.

Any idea?

Christian.

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