[Paper Review] Every multiple of 4 except 212, 364, 420, and 428 is the sum of seven cubes

Kent D. Boklan, Noam D. Elkies|ArXiv.org|Mar 26, 2009

Analytic Number Theory Research6 references5 citations

TL;DR

This paper proves that every multiple of 4, except 212, 364, 420, and 428, can be expressed as the sum of seven nonnegative cubes. Using a novel identity involving quadratic forms and primes in arithmetic progressions, the authors establish the result for all integers above $10^{18}$, and verify the remaining cases computationally, significantly advancing the seven-cubes conjecture under a congruence condition.

ABSTRACT

It is conjectured that every integer N>454 is the sum of seven nonnegative cubes. We prove the conjecture when N is a multiple of 4.

Motivation & Objective

• To prove that every multiple of 4, except four specific exceptions, is expressible as the sum of seven nonnegative cubes.
• To extend the known range of validity for the seven-cubes conjecture beyond the previously established $2.5 \times 10^{26}$ threshold.
• To provide a congruence-based result that avoids reliance on size bounds, using number-theoretic tools like prime distribution in arithmetic progressions.
• To demonstrate that the exceptional set for the seven-cubes problem is likely finite and fully characterizable under specific modular conditions.

Proposed method

• Leverages a cubic identity (Equation 4) that expresses sums of six cubes via a quadratic form $ Q = \sum_{i=1}^3 c_i X_i^2 $, where $ c_i $ are positive integers not all equal to 1.
• Applies a criterion (Proposition 1) requiring the existence of a prime $ p \equiv 2 \pmod{3} $ in the interval $ (AN^{1/3}, BN^{1/3}) $, with $ p $ dividing $ N - x_0^3 $ for a small $ x_0 $.
• Uses explicit bounds from Ramaré and Rumely (1996) on prime distribution in arithmetic progressions modulo 72 to guarantee the existence of such primes for $ N > 10^{18} $.
• Employs a finite computation of prime chains in each relevant congruence class mod 72 to verify existence of primes in the required intervals, with $ \delta = 0.01 $ ensuring sufficient density.
• Constructs a quadratic form $ 2X_1^2 + 2X_2^2 + 3X_3^2 $ that represents integers in certain arithmetic progressions and passes mod 9 representability tests.
• Verifies the remaining cases below $ 10^{18} $ computationally, confirming that only the four exceptions fail to be sums of seven nonnegative cubes.

Experimental results

Research questions

• RQ1 Can every sufficiently large multiple of 4 be written as the sum of seven nonnegative cubes, excluding a finite set of exceptions?
• RQ2 What role do quadratic forms with non-uniform coefficients play in representing integers as sums of cubes?
• RQ3 How can prime distribution in arithmetic progressions be leveraged to prove representability results in Waring-type problems?
• RQ4 To what extent can congruence conditions like $ N \equiv 0 \pmod{4} $ simplify the search for representations as sums of cubes?
• RQ5 Are the known exceptional values for the seven-cubes problem (e.g., 212, 364, 420, 428) the only ones for multiples of 4?

Key findings

• Every multiple of 4 greater than $ 10^{18} $, except 212, 364, 420, and 428, is expressible as the sum of seven nonnegative cubes.
• The proof relies on the existence of a prime $ p \equiv 2 \pmod{3} $ in the interval $ (AN^{1/3}, BN^{1/3}) $, with $ A = \beta^{-1}/\sqrt[3]{1072} $, $ B = \beta^{-1}/\sqrt[3]{904} $, for $ \beta \in \{1,5\} $, ensuring representability via the cubic identity.
• A finite computation of prime chains in each $ l \bmod 72 $ with $ l \equiv 2 \pmod{3} $ and $ \gcd(l,72)=1 $ confirms the existence of such primes for all $ N > 10^{18} $, with $ \delta = 0.01 $ ensuring the required gap ratio $ < 1.0584 $.
• The exceptional values 212, 364, 420, and 428 are confirmed not to be representable as sums of seven nonnegative cubes, and no other multiples of 4 below $ 10^{18} $ fail to be representable.
• For $ N > 10^{18} $, the representation uses only positive cubes, so every such $ N $ is a sum of seven positive cubes.
• The computation to verify the remaining cases below $ 10^{18} $, including the case $ N = 2408 $, was completed in under a minute on a standard desktop, using PARI/GP.