Let me introduce a famous story related to the number \( 1729 \). It dates back to the early 1900s and involves the British mathematician G. H. Hardy.

Remarkable Insight

While on his way to visit his friend, the Indian mathematician Srinivasa Ramanujan, who was ill, Hardy rode in a taxi with the number \( 1729 \). He mentioned to Ramanujan that it seemed to be a dull and unremarkable number to him. Ramanujan immediately replied, “Not at all. It is a very interesting number. It is the smallest number that can be expressed as the sum of two cubes in two different ways.”

This episode highlights Ramanujan’s extraordinary insight into numbers. In fact, \( 1729 \) can be expressed as follows: \[ 1729=12^3+1^3=10^3+9^3. \]

The British mathematician J. E. Littlewood once remarked about Ramanujan, saying that “every natural number was his friend.”

This story has led to \( 1729 \) being referred to as the Hardy-Ramanujan number.

Fourth Powers and Ramanujan’s Intuition

Incidentally, there is more to Hardy’s story. When he asked Ramanujan whether a similar result could be found for fourth powers, Ramanujan thought for a while and then said, “I don’t know the answer, but it would be a very large number.”

This intuition was accurate, and indeed, for fourth powers, the smallest such number is: \[ 635318675=158^4+59^4=134^4+133 ^4. \]

Beyond Conventional Understanding

Ramanujan’s genius was extraordinary. He handled mathematical formulas intuitively, as if he were breathing, and it seemed as if he was an alien who had landed on Earth and was distributing his knowledge on a whim. Moreover, it seems that he had no concept of proof. 👽️