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Multigrade Equation

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A (k,l)-multigrade equation is a Diophantine equation of the form

sum_(i=1)^ln_i^j=sum_(i=1)^lm_i^j
(1)

for j=1, ..., k, where m and n are l-vectors. Multigrade identities remain valid if a constant is added to each element of m and n (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 m={1,6,8} and n={2,4,9}:

sum_(i=1)^(3)m_i^1=sum_(i=1)^(3)n_i^1=15
(2)
sum_(i=1)^(3)m_i^2=sum_(i=1)^(3)n_i^2=101,
(3)

the (3, 4)-multigrade with m={1,5,8,12} and n={2,3,10,11}:

sum_(i=1)^(4)m_i^1=sum_(i=1)^(4)n_i^1=26
(4)
sum_(i=1)^(4)m_i^2=sum_(i=1)^(4)n_i^2=234
(5)
sum_(i=1)^(4)m_i^3=sum_(i=1)^(4)n_i^3=2366,
(6)

and the (4, 6)-multigrade with m={1,5,8,12,18,19} and n={2,3,9,13,16,20}:

sum_(i=1)^(6)m_i^1=sum_(i=1)^(6)n_i^1=63
(7)
sum_(i=1)^(6)m_i^2=sum_(i=1)^(6)n_i^2=919
(8)
sum_(i=1)^(6)m_i^3=sum_(i=1)^(6)n_i^3=15057
(9)
sum_(i=1)^(6)m_i^4=sum_(i=1)^(6)n_i^4=260755
(10)

(Madachy 1979).

A spectacular example with k=9 and l=10 is given by n={+/-12,+/-11881,+/-20231,+/-20885,+/-23738} and m={+/-436,+/-11857,+/-20449,+/-20667,+/-23750} (Guy 1994), which has sums

sum_(i=1)^(9)m_i^1=sum_(i=1)^(9)n_i^1=0
(11)
sum_(i=1)^(9)m_i^2=sum_(i=1)^(9)n_i^2=3100255070
(12)
sum_(i=1)^(9)m_i^3=sum_(i=1)^(9)n_i^3=0
(13)
sum_(i=1)^(9)m_i^4=sum_(i=1)^(9)n_i^4=1390452894778220678
(14)
sum_(i=1)^(9)m_i^5=sum_(i=1)^(9)n_i^5=0
(15)
sum_(i=1)^(9)m_i^6=sum_(i=1)^(9)n_i^6=666573454337853049941719510
(16)
sum_(i=1)^(9)m_i^7=sum_(i=1)^(9)n_i^7=0
(17)
sum_(i=1)^(9)m_i^8=sum_(i=1)^(9)n_i^8=330958142560259813821203262692838598
(18)
sum_(i=1)^(9)m_i^9=sum_(i=1)^(9)n_i^9=0.
(19)

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

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

a_1^4+a_2^4+a_3^4=2a_4^m
(20)

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

(a^2+ab)^r+(a^2-ab)^r+(b^2+ab)^r+(b^2-ab)^r=2(p^2+q^2)^(kr)
(21)

with r=1, 2, 3, for any positive integer k, and where

a=1/2[(p-qi)^k+(p+qi)^k]
(22)
b=1/2i[(p-qi)^k-(p+qi)^k]
(23)

and the one for fifth powers

[(a+c)^n+(b+c)^n+(a+b+c)^n+(-a-b+c)^n+(-b+c)^n+(-a+c)^n](2/3)^n=2(p^2+pq+q^2)^(hn)
(24)

for n=1, 3, 5, any positive integer h, and where

a=(omega(p-qomega)^(2h)-(p-qomega^2)^(2h))/(omega-1)
(25)
b=((p-qomega)^(2h)-(p-qomega^2)^(2h))/(omega(omega-1))
(26)
c=1/2(p^2+pq+q^2)^h
(27)

with omega a complex cube root of unity and a and b for both cases rational for arbitrary rationals p and q.

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 k.4.4,

(ax+v_1y)^k+(bx-v_1y)^k+(cx-v_2y)^k+(dx+v_2y)^k =(ax-v_1y)^k+(bx+v_1y)^k+(cx+v_2y)^k+(dx-v_2y)^k (ax^2-v_1xy+bwy^2)^k+(bx^2+v_1xy+awy^2)^k+(cx^2+v_2xy+dwy^2)^k+(dx^2-v_2xy+cwy^2)^k =(a^k+b^k+c^k+d^k)(x^2+wy^2)^k
(28)

for k=1, 3, (v_1,v_2)=(c^2-d^2,a^2-b^2), and w=(a+b) or (c+d) for arbitrary a, b, c, d, x, and y.

For fourth powers with k.3.3,

(ax+v_1y)^k+(bx-v_2y)^k+(cx-v_3y)^k =(ax-v_1y)^k+(bx+v_2y)^k+(cx+v_3y)^k +(ax^2+2v_1xy-3ay^2)^k+(bx^2-2v_2xy-3by^2)^k+(cx^2-2v_3xy-3cy^2)^k=(a^k+b^k+c^k)(x^2+3y^2)^k
(29)

for k=2, 4, (v_1,v_2,v_3,c)=(a+2b,2a+b,a-b,a+b), for arbitrary a, b, x, y.

For fifth powers with k.6.6,

(a_1x+v_1y)^k+(a_2x-v_2y)^k+(a_3x+v_3y)^k+(a_4x-v_3y)^k+(a_5x+v_2y)^k+(a_6x-v_1y)^k =(a_1x-v_1y)^k+(a_2x+v_2y)^k+(a_3x-v_3y)^k+(a_4x+v_3y)^k+(a_5x-v_2y)^k+(a_6x+v_1y)^k =(a_1x^2+2v_1xy+3a_6y^2)^k+(a_2x^2-2v_2xy+3a_5y^2)^k+(a_3x^2+2v_3xy+3a_4y^2)^k+(a_4x^2-2v_3xy+3a_3y^2)^k+(a_5x^2+2v_2xy+3a_2y^2)^k+(a_6x^2-2v_1xy+3a_1y^2)^k =(a_1^k+a_2^k+a_3^k+a_4^k+a_5^k+a_6^k)(x^2+3y^2)^k
(30)

for k=1, 2, 3, 4, 5, (a_1,a_2,a_3,a_4,a_5,a_6)=(a+c,b+c,-a-b+c,a+b+c,-b+c,-a+c), (v_1,v_2,v_3)=(a+2b,2a+b,a-b) (which are the same v_i for fourth powers) for arbitrary a, b, c, x, y and one for seventh powers that uses sqrt(2).

For seventh powers with k.8.8,

(a_1x+v_1y)^k+(a_2x+v_2y)^k+(a_3x+v_3y)^k+(a_4x+v_4y)^k +(a_5x-v_4y)^k+(a_6x-v_3y)^k+(a_7x-v_2y)^k+(a_8x-v_1y)^k =(a_1x-v_1y)^k+(a_2x-v_2y)^k +(a_3x-v_3y)^k+(a_4x-v_4y)^k +(a_5x+v_4y)^k+(a_6x+v_3y)^k+(a_7x+v_2y)^k+(a_8x+v_1y)^k +(a_1x^2+v_1xy+a_8y^2)^k+(a_2x^2+v_2xy+a_7y^2)^k +(a_3x^2+v_3xy+a_6y^2)^k+(a_4x^2+v_4xy+a_5y^2)^k +(a_5x^2-v_4xy+a_4y^2)^k+(a_6x^2-v_3xy+a_3y^2)^k +(a_7x^2-v_2xy+a_2y^2)^k+(a_8x^2-v_1xy+a_1y^2)^k =(a_1^k+a_2^k+a_3^k+a_4^k+a_5^k+a_6^k+a_7^k+a_8^k)(x^2+y^2)^k
(31)

for k=1 to 7, (a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8)=(sqrt(2)a+c,sqrt(2)b+c,-a+b+c,-a-b+c,a+b+c,a-b+c,-sqrt(2)b+c,-sqrt(2)+c), (v_1,v_2,v_3,v_4)=(2sqrt(2)b,-2sqrt(2)a,-2(a+b),2(a-b)), for arbitrary, a, b, c, x, y (Piezas 2006).

A multigrade 5-parameter binary quadratic form identity exists for k.4.4 with k=1, 2, 3, 5. Given arbitrary variables a, b, c, x, y and defining u=a^2-b^2 and v=b^2-c^2, then

[(-a+b+c)x^2+2(cu-bv)xy-(a+b+c)uvy^2]^k+[(a-b+c)x^2+2(cu+bv)xy+(a+b-c)uvy^2]^k+[(a+b-c)x^2+2(-cu-bv)xy+(a-b+c)uvy^2]^k+[-(a+b+c)x^2+2(-cu+bv)xy+(-a+b+c)uvy^2]^k-[-(a+b+c)x^2+2(-bu+av)xy+(a+b-c)uvy^2]^k-[(a+b-c)x^2+2(bu-av)xy-(a+b+c)uvy^2]^k-[(a-b+c)x^2+2(-bu-av)xy+(-a+b+c)uvy^2]^k-[(-a+b+c)x^2+2(bu+av)xy+(a-b+c)uvy^2]^k=0
(32)

for k=1, 2, 3, 5 (T. Piezas, pers. comm., Apr. 27, 2006).

Chernick (1937) gave a multigrade binary quadratic form parametrization to k.4.4 for k=2, 4, 6 given by

(5m^2+9mn+10n^2)^k+(m^2-13mn-6n^2)^k+(7m^2-5mn-8n^2)^k+(9m^2+7mn-4n^2)^k =(9m^2+5mn+4n^2)^k+(m^2+15mn+8n^2)^k+(5m^2-7mn-10n^2)^k+(7m^2+5mn-6n^2)^k,
(33)

an equation which depends on finding solutions to 4a^2+ab+b^2=7c^2.

Sinha (1966ab) gave a multigrade binary quadratic form parametrization to k.5.5 for k=1, 3, 5, 7 given by

(-7m^2+62mn-30n^2)^k+(7m^2+38mn-50n^2)^k+(5m^2-8mn-22n^2)^k+(19m^2-32mn-42n^2)^k+(-19m^2+36mn-62n^2)^k =(-9m^2+66mn-42n^2)^k+(5m^2+42mn-62n^2)^k+(-21m^2+38mn-22n^2)^k+(9m^2-14mn-50n^2)^k+(21m^2-36mn-30n^2)^k
(34)

which depended on solving the system a_1^j+a_2^j+a_3^j=b_1^j+b_2^j+b_3^j for j=2 and 4 with a_i and b_i satisfying certain other conditions.

Sinha (1966ab), using a result of Letac, also gave a multigrade parametrization to k.5.5 for k=1, 2, 4, 6, 8 given by

(a-r)^k+(a+r)^k+(3b-t)^k+(3b+t)^k+(4a)^k =(b-t)^k+(b+t)^k+(3a-r)^k+(3a+r)^k+(4b)^k,
(35)

where a^2+12b^2=r^2 and 12a^2+b^2=t^2. One nontrivial solution can be given by a=109, b=11869/2, and Sinha and Smyth proved in 1990 that there are an infinite number of distinct nontrivial solutions.

See also

Diophantine Equation, Prouhet-Tarry-Escott Problem

Portions of this entry contributed by Tito Piezas III

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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 A_1, A_2, A_3, A_4, A_5=^kB_1, B_2, B_3, B_4, B_5 (k=1, 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." http://www.geocities.com/titus_piezas/Ramfifth.html.Piezas, T. "Binary Quadratic Forms as Equal Sums of Like Powers." http://www.geocities.com/titus_piezas/Binary_quad.html.Rivera, C. "Problems & Puzzles: Puzzle 065-Multigrade Relations." http://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.

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Multigrade Equation

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Piezas, Tito III and Weisstein, Eric W. "Multigrade Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MultigradeEquation.html

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