Ramanujan Surprises Again (2015)

An interesting coincidence: it was recently (2019) discovered that the fastest way to multiply two n-bit integers, in time O(n log n), involves 1729-dimensional Fourier transforms: <a href="https:&#x2F;&#x2F;hal.archives-ouvertes.fr&#x2F;hal-02070778" rel="nofollow">https:&#x2F;&#x2F;hal.archives-ouvertes.fr&#x2F;hal-02070778</a>. It is quite surprising that the asymptotically best way to perform such an elementary operation should be tied to Ramanujan’s famous taxicab number.<p>(Technically, it works for any number of dimensions &gt;= 1729, but the proof fails for dimensions less than that. Future work might bring the bound down, or better explain why that bound is necessary.)

He credited his work to his family goddess. From wikipedia:<p>&quot;A deeply religious Hindu, Ramanujan credited his substantial mathematical capacities to divinity, and said the mathematical knowledge he displayed was revealed to him by his family goddess. &quot;An equation for me has no meaning,&quot; he once said, &quot;unless it expresses a thought of God.&quot;&quot;

Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan&#x27;s innate calculating capability. It&#x27;s interesting to find out that the real coincidence here is that Hardy rode in a taxicab whose number had happened to show up in Ramanujan&#x27;s investigations of Fermat&#x27;s last theorem.

The taxi cab story is easily a top-5 math story, and is quintessential Ramanujan.<p>Has there been a genius of his kind since? Maybe Terry Tao, but his work also lacks the ease and lack of machinery that Ramanujan had. Truly amazing.

Discussed at the time: <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=10518452" rel="nofollow">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=10518452</a>

I always found Ramanujan very intriguing. He operates on a dimension that is unknown to most of us. Makes me wonder if he is a great yogi or a time traveler.

Fantastic article that explains the math (and physics) very clearly.

Interesting read, The title should have 2015 in it though.

Ramanujan also claimed 1 + 2 + 3 + ... = -1&#x2F;12.<p>How does that work? Who can explain this to me?

Don&#x27;t we think that the credit for the number 1729 should belong to Hardy, for he took the cab and mentioned that number to Ramanujan. Of course, Ramanujan could see beauty in every number and would have produced something equally beautiful for some other number Hardy could utter.