I. Some General Conjectures and ProblemsII. Some TheoremsIII. Fifth Powers5.1 Four terms 5.2 Six terms 5.3 Seven terms 5.4 Eight terms 5.5 Ten terms 5.6 Twelve terms
Most of the identities I found for kth powers involve
multigrade "Equal Sums of Like Powers (ESLP)" (multigrade being the case valid for several exponents). For numerical examples that solve the "Prouhet-Tarry-Escott Problem (PTE)" in certain cases, see Chen Shuwen's site. For the definitive site on the present status of ESLP, kindly see Jean-Charles Meyrignac's "Computing Minimal Equal Sums of Like Powers".
1.
This was formalized by R. Ekl’s 1998 “
2.
formalized by J. Meyrignac when he started EulerNet in 1999. Thus this specifically demands that, for example, (4,1,3), (5,1,4), (6,1,5) have a soln. While it is indeed true for the first two, for the third, no soln is known even for (6,1,6) so only time will tell if the conjecture will hold.
3.
Assorted Identities.)
For simplicity, we will designate the multi-grade system a
[a
where
"For
In others words, n ≤ 11 other than n = 10. The next theorems allow transformations to be done on them.
"If [a
For example, using the solution [1, 5, 6] = [2, 3, 7], k = 1,2, then for any x,
(x+1)
[1, 5, 8, 12] = [2, 3, 10, 11], k = 1,2,3
since (x+1)+(x+5)+(x+8)+(x+12) = 0 gives x = -26/4, then,
[-11, -3, 3, 11] = [-9, -7, 7, 9], k = 1,2,3
after removing common factors. It should be pointed out that not all become
[1, 75, 87, 205] = [15, 37, 115, 201], k = 1,2,3, yields,
[-91, -17, -5, 113] = [-77, -55, 23, 109], k = 1,2,3,5
Naturally enough, these are called
"If [a
Ex. [1, 5, 6] = [2, 3, 7], k = 1,2 gives
[1, 5, 6, x+2, x+3, x+7] = [2, 3, 7, x+1, x+5, x+6], k = 1,2,3
As expected, the number of terms doubles but with carefully chosen terms, this doubling can be prevented since some terms may appear on both sides of the equation and just cancel out. For example, this one by Tarry,
[1, 5, 10, 16, 27, 28, 38, 39] = [2, 3, 13, 14, 25, 31, 36, 40], k = 1,2,…6
using the theorem starting with x = 11 found the first ideal soln with degree n=7,
[1, 5, 10, 24, 28, 42, 47, 51] = [2, 3, 12, 21, 31, 40, 49, 50], k = 1,2,…7
since eight terms on either side cancel out. This author has found a generalization of this particular example to be discussed on the section on
"If [a
Or alternatively, since we are dealing with odd powers and can move all terms to the LHS,
"If [a
Ex. [0, 7, 8, -1, -5, -9] = 0, k = 1,3 gives,
[x+0, x+7, x+8, x-1, x-5, x-9] = [x-0, x-7, x-8, x+1, x+5, x+9], k = 1,2,3,4
Note that appropriate pairs of terms taken from
[0, 3] = [1, 2], k = 1 [0, 7, 8] = [1, 5, 9], k = 1,3[0, 24, 33, 51] = [7, 13, 38, 50], k = 1,3,5 [0, 34, 58, 82, 98] = [13, 16, 69, 75, 99], k = 1,3,5,7 (A. Letac, 1942)
with the last found pre-computer searching. Parametric solns are known for all except the case k = 1,3,5,7. (
"If [a
Ex. [1, 9, 10] = [5, 6, 11], k = 2,4 gives,
[x+1, x+9, x+10, x-1, x-9, x-10] = [x+5, x+6, x+11, x-5, x-6, x-11], k = 1,2,3,4,5
If m = n+1, this automatically gives an ideal soln of degree 2n+1. Note that pairs of terms now from the
[1, 7] = [5, 5], k = 2 [1, 9, 10] = [5, 6, 11], k = 2,4 [2, 16, 21, 25] = [5, 14, 23, 24], k = 2,4,6 (G. Tarry, 1913) [71, 131, 180, 307, 308] = [99, 100, 188, 301, 313], k = 2,4,6,8 (Borwein, Lisonek, Percival) [22, 61, 86, 127, 140, 151] = [35, 47, 94, 121, 146, 148], k = 2,4,6,8,10 (Kuosa, Meyrignac, and Shuwen)
Parameterizations are known for all these cases. For k = 2,4,6,8, Letac earlier gave a method using an elliptic curve though it results in relatively large numbers. For k = 2,4,6,8,10, after Choudhry and Wroblewski recently found also an elliptic curve (in 2008), it is now known that this system has an infinite number of solns. In general, Theorems 4 and 5 yield what are called
"If [a
Gloden's theorem is equivalent to reducing ideal solns to its
[1, 75, 87, 205] = [15, 37, 115, 201], k = 1,2,3
we get,
[-91, -17, -5, 113] = [-77, -55, 23, 109], k = 1,2,3,5
where sum of terms Σa
(Update, 8/7/09): For an extensive collection of theorems involving equal sums of like powers, see Part 3 of |