Multigrade Equation

A -multigrade equation is a Diophantine equation of the form

(1)

for , ..., , where  and  are -vectors. Multigrade identities remain valid if a constant is added to each element of  and  (Madachy 1979), so multigrades can always be put in a form where the minimum component of one of the vectors is 1.

Moessner and Gloden (1944) give a bevy of multigrade equations. Small-order examples are the (2, 3)-multigrade with  and :

			(2)
			(3)

the (3, 4)-multigrade with  and :

			(4)
			(5)
			(6)

and the (4, 6)-multigrade with  and :

			(7)
			(8)
			(9)
			(10)

(Madachy 1979).

A spectacular example with  and  is given by  and  (Guy 1994), which has sums

			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)

Rivera considers multigrade equations involving primes, consecutive primes, etc.

Analogous multigrade identities to Ramanujan's fourth power identity of form

(20)

can also be given for third and fifth powers, the former being

(21)

with , 2, 3, for any positive integer , and where

			(22)
			(23)

and the one for fifth powers

(24)

for , 3, 5, any positive integer , and where

			(25)
			(26)
			(27)

with  a complex cube root of unity and  and  for both cases rational for arbitrary rationals  and .

Multigrade sum-product identities as binary quadratic forms also exist for third, fourth, fifth powers. These are the second of the following pairs.

For third powers with ,

(28)

for , 3, , and  or  for arbitrary , , , , , and .

For fourth powers with ,

(29)

for , 4, , for arbitrary , , , .

For fifth powers with ,

(30)

for , 2, 3, 4, 5, ,  (which are the same  for fourth powers) for arbitrary , , , ,  and one for seventh powers that uses .

For seventh powers with ,

(31)

for  to 7, , , for arbitrary, , , , ,  (Piezas 2006).

A multigrade 5-parameter binary quadratic form identity exists for  with , 2, 3, 5. Given arbitrary variables , , , ,  and defining  and , then

(32)

for , 2, 3, 5 (T. Piezas, pers. comm., Apr. 27, 2006).

Chernick (1937) gave a multigrade binary quadratic form parametrization to  for , 4, 6 given by

(33)

an equation which depends on finding solutions to .

Sinha (1966ab) gave a multigrade binary quadratic form parametrization to  for , 3, 5, 7 given by

(34)

which depended on solving the system  for  and 4 with  and  satisfying certain other conditions.

Sinha (1966ab), using a result of Letac, also gave a multigrade parametrization to  for , 2, 4, 6, 8 given by

(35)

where  and . One nontrivial solution can be given by , , and Sinha and Smyth proved in 1990 that there are an infinite number of distinct nontrivial solutions.

REFERENCES:

Chernick, J. "Ideal Solutions of the Tarry-Escott Problem." Amer. Math. Monthly 44, 62600633, 1937.

Gloden, A. Mehrgeradige Gleichungen. Groningen, Netherlands: Noordhoff, 1944.

Gloden, A. "Sur la multigrade , , , , , , , ,  (, 3, 5, 7)." Revista Euclides 8, 383-384, 1948.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 143, 1994.

Kraitchik, M. "Multigrade." §3.10 in Mathematical Recreations. New York: W. W. Norton, p. 79, 1942.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 171-173, 1979.

Moessner, A. and Gloden, A. "Einige Zahlentheoretische Untersuchungen und Resultate." Bull. Sci. École Polytech. de Timisoara 11, 196-219, 1944.

Piezas, T. "Ramanujan and Fifth Power Identities." https://www.geocities.com/titus_piezas/Ramfifth.html.

Piezas, T. "Binary Quadratic Forms as Equal Sums of Like Powers." https://www.geocities.com/titus_piezas/Binary_quad.html.

Rivera, C. "Problems & Puzzles: Puzzle 065-Multigrade Relations." https://www.primepuzzles.net/puzzles/puzz_065.htm.

Sinha, T. "On the Tarry-Escott Problem." Amer. Math. Monthly 73, 280-285, 1966a.

Sinha, T. "Some System of Diophantine Equations of the Tarry-Escott Type." J. Indian Math. Soc. 30, 15-25, 1966b.