Hardy-Ramanujan Number/Examples/6,963,472,309,248

Theorem

The $4$th Hardy-Ramanujan number $\operatorname {Ta} \left({4}\right)$ is $6 \, 963 \, 472 \, 309 \, 248$:

\(\ds 6 \, 963 \, 472 \, 309 \, 248\)

\(=\)

\(\ds 2421^3 + 19 \, 083^3\)

\(\ds \)

\(=\)

\(\ds 5436^3 + 18 \, 948^3\)

\(\ds \)

\(=\)

\(\ds 10 \, 200^3 + 18 \, 072^3\)

\(\ds \)

\(=\)

\(\ds 13 \, 322^3 + 16 \, 630^3\)

Historical Note

It is widely reported that the $4$th Hardy-Ramanujan number $\map {\operatorname {Ta} } 4$ was discovered by E. Rosenstiel, J.A. Dardis and C.R. Rosenstiel in $1991$.

However, David Wells reports in his Curious and Interesting Numbers, 2nd ed. of $1997$ that this result can be found in the Numbers Count column of Personal Computer World, November $1989$.

Proof

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Sources

• July 1991: E. Rosenstiel, J.A. Dardis and C.R. Rosenstiel: The Four Least Solutions in Distinct Positive Integers of the Diophantine Equation $s = x^3 + y^3 = z^3 + w^3 = u^3 + v^3 = m^3 + n^3$ (The Institute of Mathematics and its Applications Bulletin Vol. 27: pp. 155 – 157)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6,963,472,309,248$