A359314 Three-column table T(n,k) read by rows where the elements in the pair of two adjacent rows, starting with the odd-indexed row T(2j-1,k) and followed by the even-indexed one T(2j,k), are such that they are not multiples of the elements presented in the previous rows and that Sum_{k=1..3} T(2j-1,k)^2 = Sum_{k=1..3} T(2j,k)^2 and Sum_{k=1..3} T(2j-1,k)^6 = Sum_{k=1..3} T(2j,k)^6 for j > 0 and k = 1, 2, 3.

3, 19, 22, 10, 15, 23, 15, 52, 65, 36, 37, 67, 23, 54, 73, 33, 47, 74, 3, 55, 80, 32, 43, 81, 11, 65, 78, 37, 50, 81

Offset: 1

Author

Alexander R. Povolotsky, Dec 25 2022

Keywords

• nonn
• tabf
• more

Comments

It was found empirically (via computer calculations) that for integers a, b, c, d, e and f satisfying a^6 + b^6 + c^6 = d^6 + e^6 + f^6, it is also most likely to be true that a^2 + b^2 + c^2 = d^2 + e^2 + f^2.

Such cases are presented in this sequence where

a = T(2j-1,1), b = T(2j-1,2) c = T(2j-1,3) and

d = T(2j,1), e = T(2j,2), f = T(2j,3).

There currently exists no formula to calculate terms of this sequence -- they have to be found via trial and test (computer) calculations.

Each row consists of 3 columns.

The table starts with the rows which have the smallest sums of squares of elements (such sums also correspond to the smallest sums of the same 6th powers of the same elements) -- see the EXAMPLE section. The terms in each row are presented in ascending order.

Examples

			Table begins:
      k=1 k=2 k=3 SquaresSum 6thPowersSum
n=1:   3, 19, 22;   854         160426514
n=2:  10, 15, 23;   854         160426514
n=3:  15, 52, 65;  7154       95200890914
n=4:  36, 37, 67;  7154       95200890914
n=5:  23, 54, 73;  8774      176277173474
n=6:  33, 47, 74;  8774      176277173474
n=7:   3, 55, 80;  9434      289824641354
n=8:  32, 43, 81;  9434      289824641354
n=9:  11, 65, 78; 10430      300620262890
n=10: 37, 50, 81; 10430      300620262890
...
The elements of the row n=1: 3, 19, 22 and the elements of the row n=2: 10, 15, 23 are such that 3^2 + 19^2 + 22^2 = 10^2 + 15^2 + 23^2 and 3^6 + 19^6 + 22^6 = 10^6 + 15^6 + 23^6.
		

References

• R. K. Guy, Unsolved problems in Number theory, chapter D, section D1, page 213.

Links

• Simcha Brudno, Triples of 6th powers with equal sums, Mathematics of Computation, vol. 30, nb. 135, July 1976.
• Bernard Montaron, How can we prove that if six natural integers are such that p^6+q^6+r^6=u^6+v^6+w^6 then p^2+q^2+r^2=u^2+v^2+w^2?, Quora.