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Numbers not of the form a^2 + b^3 + c^4 + d^5 for a,b,c,d >= 0.

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`%I #18 May 15 2022 12:32:02
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`%S 15,23,55,62,71,471,478,510,646,806,839,879,939,1023,1063,1287,2127,
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`%T 5135,6811,7499,9191,26471
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`%N Numbers not of the form a^2 + b^3 + c^4 + d^5 for a,b,c,d >= 0.
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`%C It is conjectured that this list is complete.
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`%C Comments from Richard C. Schroeppel: (Start)
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`%C "I can't put my hand on the book, but I think Vaughn has shown that every sufficiently large number is of the form a^2+b^3+c^5.
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`%C The key is that the sum 1/2 + 1/3 + 1/5 = 31/30 > 1, so the expected number of representations for a number N is, on average, K * N^1/30.
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`%C K is some mess of gamma(fractions). [It's also necessary to check that there's no modular exclusion.] I once tried to estimate how large an N was "sufficiently large" and if I remember correctly got around 10^60.
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`%C The results for a^2+b^3+c^4+d^5 and a^2+b^3+c^4+d^5+e^6 would follow immediately, although proving an upper bound is a big question. I'm not even sure if Vaughn's proof is constructive.
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`%C _Christopher Landauer_ worked on the 2...6 problem long long ago; I think his program got up to a million or so.
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`%C There was a paper in Math. Comp. about five years ago, about the problem of 4 cubes. Beeler & I tried to find the empirical last-unrepresentable number, but couldn't reach it. The Math. Comp. paper found a likely candidate. It was around 10^13 or 14." (End)
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`%C No other terms < 10^8. - _T. D. Noe_, Mar 10 2008
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`%t Complement[Range[10^6], Flatten[Table[a^2 + b^3 + c^4 + d^5, {a, 0, 1000}, {b, 0, 100}, {c, 0, 31}, {d, 0, 15}]]] (* _Robert G. Wilson v_, Oct 19 2005 *)
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`%Y Cf. A135911 (number of 4-tuples (x, y, z, t) of nonnegative integers such that x^2+y^3+z^4+t^5 = n).
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`%K nonn,fini
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`%O 1,1
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`%A _David W. Wilson_, Oct 19 2005
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