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We call a constellation **ideal** if there are n different powers and no more than n+1 powers on each side of the equation. Here are the ideal constellations with distinct terms and the smallest known sums:

(1,2): 1^{n} + 2^{n} + 6^{n} =
4^{n} + 5^{n} (EF)

(1,3): 1^{n} + 5^{n} + 9^{n} =
7^{n} + 8^{n} (EF, BC, PF)

(1,4): 3^{n} + 25^{n} + 38^{n} =
7^{n} + 20^{n} + 39^{n} (CS)

(1,5): 3^{n} + 54^{n} + 62^{n} =
24^{n} + 28^{n} + 67^{n} (LL)

(2,3): 37^{n} + 62^{n} =
21^{n} + 26^{n} + 64^{n} (EF)

(2,4): 5^{n} + 6^{n} + 11^{n} =
1^{n} + 9^{n} + 10^{n} (BC)

(2,6): 3^{n} + 19^{n} + 22^{n} =
10^{n} + 15^{n} + 23^{n} (LL)

(1,2,3): 4^{n} + 7^{n} + 11^{n} =
1^{n} + 2^{n} + 9^{n} + 10^{n} (BC, PF)

(1,2,4): 2^{n} + 7^{n} + 11^{n} + 15^{n} =
3^{n} + 5^{n} + 13^{n} + 14^{n} (CS)

(1,2,5): 1^{n} + 28^{n} + 39^{n} + 58^{n} =
8^{n} + 14^{n} + 51^{n} + 53^{n} (CS)

(1,2,6): 7^{n} + 16^{n} + 25^{n} + 30^{n} =
8^{n} + 14^{n} + 27^{n} + 29^{n} (CS)

(1,3,4): 3^{n} + 140^{n} + 149^{n} + 252^{n} =
50^{n} + 54^{n} + 201^{n} + 239^{n} (CS)

(1,3,5): 6^{n} + 16^{n} + 18^{n} + 24^{n} =
7^{n} + 13^{n} + 21^{n} + 23^{n} (CS)

(1,3,7): 184^{n} + 443^{n} + 556^{n} + 698^{n} =
230^{n} + 353^{n} + 625^{n} + 673^{n} (CS)

(1,2,3,4): 1^{n} + 2^{n} + 10^{n} + 14^{n} + 18^{n} =
4^{n} + 8^{n} + 16^{n} + 17^{n} (PF)

(1,2,3,5): 1^{n} + 8^{n} + 13^{n} + 24^{n} + 27^{n} =
3^{n} + 4^{n} + 17^{n} + 21^{n} + 28^{n} (CS)

(1,2,3,6): 7^{n} + 18^{n} + 55^{n} + 69^{n} + 81^{n} =
9^{n} + 15^{n} + 61^{n} + 63^{n} + 82^{n} (CS)

(1,2,4,6): 1^{n} + 19^{n} + 22^{n} + 37^{n} + 42^{n} =
2^{n} + 14^{n} + 29^{n} + 33^{n} + 43^{n} (CS)

(1,2,3,4,5): 1^{n} + 2^{n} + 10^{n} + 12^{n} + 20^{n} + 21^{n} =
5^{n} + 6^{n} + 16^{n} + 17^{n} + 22^{n} (PF)

(1,2,3,4,5,6): 14^{n} + 16^{n} + 45^{n} + 54^{n} + 73^{n} + 83^{n} =
3^{n} + 5^{n} + 28^{n} + 34^{n} + 65^{n} + 66^{n} + 84^{n} (PF)

(-1,1): 4^{n} + 10^{n} + 12^{n} =
5^{n} + 6^{n} + 15^{n} (CS)

(-1,2): 35^{n} + 65^{n} + 84^{n} =
39^{n} + 52^{n} + 91^{n} (CS)

(-2,1): 60^{n} + 105^{n} + 140^{n} =
65^{n} + 84^{n} + 156^{n} (CS)

Here are the ideal constellations with the smallest known sums, if different than above:

(1,2): 3^{n} + 3^{n} =
1^{n} + 1^{n} + 4^{n} (EF, PF)

(2,4): 3^{n} + 5^{n} + 8^{n} =
7^{n} + 7^{n} (EF)

(1,2,3): 1^{n} + 1^{n} + 6^{n} + 6^{n} =
3^{n} + 4^{n} + 7^{n} (PF)

(1,2,4,6): 3^{n} + 7^{n} + 10^{n} + 16^{n} + 16^{n} =
4^{n} + 5^{n} + 12^{n} + 14^{n} + 17^{n} (CS)

(1,2,3,4,5): 3^{n} + 5^{n} + 11^{n} + 13^{n} + 16^{n}=
1^{n} + 1^{n} + 8^{n} + 8^{n} + 15^{n} + 15^{n} (GT)

Boris Bukh tells me that the problem of estimating the number of (1,2,...,k) constellations, where the number of summands on the left and on the right is the same, is at the heart of the Vinogradov's mean-value method for bounding exponential sums. He suggests Titchmarsh's "Theory of Riemann zeta-function", chapter VI for discussion of the method, and known results.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 3/8/04.