0021d: Article 14 (Euler Four-Square Variants)

An Infinite Family of Four-Square and Eight-Square Identities

By Tito Piezas III

Abstract: An infinite family of 8-square identities (which can be specialized into a 4-square) will be given.

I. Introduction

II. A Family of 4-Square Identities

III. A Family of 8-Square Identities

I. Introduction

In the previous article, Pfister’s 16-Square Identity, aside from giving a non-bilinear 16-square identity, a variation of Euler’s Four-Square Identity with only two bilinear z_i was also mentioned, namely,

One can ask a reasonable question: How many four-square identities are there? It can be shown that there are an infinite number of four-square and eight-square identities where not all z_i are bilinear.

II. A Family of 4-Square Identities

In contrast to the Pfister 4-square, the one below generally has only one bilinear z_i,

for arbitrary n. Of course, for n = x₄, then k = 0, all the z_i become bilinear, hence the Euler 4-square can be considered a special case of this family. The {a_i, b_i, c_i} also satisfy the nice relations,

a₁² + b₁² + c₁² = x₁² + x₂² + x₄²

a₂² + b₂² + c₂² = x₁² + x₃² + x₄²

a₃² + b₃² + c₃² = x₂² + x₃² + x₄²

and,

a₁² + a₂² + a₃² = x₂² + x₃² + x₄²

b₁² + b₂² + b₃² = x₁² + x₃² + x₄²

c₁² + c₂² + c₃² = x₁² + x₂² + x₄²

If we include the leading coefficients of the first three z_i as {a₀, b₀, c₀} = {x₁, x₂, x₃}, then the last three relations simplify as,

Σ a_i² = Σ b_i² = Σ c_i² = Σ x_j²

However, this 4-square family in turn is just a special case of an 8-square family.

III. A Family of 8-Square Identities

An 8-square identity with four bilinear z_i was given in Pfister’s 16-Square Identity. But we can also give one with five bilinear z_i. First, define the following variables,

where,

m = x₄-n

and the other variables to be defined later. Also,

For the latter set, note that, for any value of the variables, then,

hence, they are just three versions of the Euler 4-Square. And where do the {a_i, b_i, c_i} appear? In an 8-square identity, namely,

and we can now define {p_i, q_i, r_i, s_i, t} in terms of the x_i and n as,

for t = 0, and arbitrary n. (The variable t was just a place-holder.) With these definitions, it can be seen that,

and the {a_i, b_i, c_i} will also satisfy the additional nice relations,

a₁² + b₁² + c₁² = x₁² + x₂² + x₄²

a₂² + b₂² + c₂² = x₁² + x₃² + x₄²

a₃² + b₃² + c₃² = x₂² + x₃² + x₄²

a₄² + b₄² + c₄² = x₅² + x₆² + x₇²

a₅² + b₅² + c₅² = x₅² + x₆² + x₈²

a₆² + b₆² + c₆² = x₅² + x₇² + x₈²

a₇² + b₇² + c₇² = x₆² + x₇² + x₈²

as well as,

a₁² + a₂² + a₃² + a₄² + a₅² + a₆² + a₇² = x₂² + x₃² + x₄² + x₅² + x₆² + x₇² + x₈²

b₁² + b₂² + b₃² + b₄² + b₅² + b₆² + b₇² = x₁² + x₃² + x₄² + x₅² + x₆² + x₇² + x₈²

c₁² + c₂² + c₃² + c₄² + c₅² + c₆² + c₇² = x₁² + x₂² + x₄² + x₅² + x₆² + x₇² + x₈²

Again, if we include the leading coefficients as {a₀, b₀, c₀} = {x₁, x₂, x₃}, then the last three relations simplify as,

Σ a_i² = Σ b_i² = Σ c_i² = Σ x_k²

While n is arbitrary, there is no value n such that all the z_i become bilinear, hence this 8-square family cannot reduce to the Degen-Graves 8-square. However, if we set {x_i, y_i} = 0 for all i > 4, then it does reduce to the 4-square family in the previous section where, as was pointed out already, the special case n = x₄ further reduces it to the Euler 4-square. There is in fact an 8-square family that can reduce to the Degen-Graves, but its variables are too tediously complicated for this article. The one given suffices to prove there is an infinite number of 8-square identities.

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You can email author at tpiezas@gmail.com.

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