Taxicab number

In mathematics, the - te Taxicab number is defined as the smallest (natural ) number that can be represented as the sum of two cubes in different ways. Godfrey Harold Hardy and EM Wright in 1954 proved that there is a Taxicab number for each natural number. The proof, however, says nothing about the occurrence of these numbers so they can be used only with large ( computer-aided ) effort found.

It was named after a famous anecdote of Hardy. He visited Ramanujan at the bedside and mentioned that he had come with a taxi number 1729, which gave Hardy a very uninteresting number. Ramanujan found this not by Hardy outlined the characteristics mentioned above.

Known Taxicab numbers

Currently, only the first six Taxicab numbers known ( sequence A011541 in OEIS ):

Discovery history

Ta ( 2) = 1729 by virtue of the above anecdote also known as the Hardy - Ramanujan number; it was published in 1657 by Bernard de Bessy Frénicle.

TA (3) = 87539319 was discovered by John Leech 1957.

Ta ( 4) was found in 1991 by the amateur number theorist E. Rosenstiel

Ta ( 5) is owed since 1999 David W. Wilson .. Regardless, they also found a few months later, Daniel Bernstein.

Ta (6) was discovered in 2003. Previously, in 1998, Daniel Bernstein had given an upper limit.

For more Taxicab numbers and upper bounds are known.

Generalized Taxicab number

As a generalized Taxicab numbers are called a modification of the ordinary Taxicab numbers. The definition is:

For and it is the "ordinary" Taxicab numbers.

Leonhard Euler showed that applies

An unsolved problems of mathematics is an existence theorem for other values ​​as and. For these values, even with computer assistance no solutions were found. This problem is related to the Euler 's conjecture, a generalization of the Great Fermat's theorem.

Literature and links

• Joseph Silverman: Taxicabs and Sums of Two Cubes. In: American Mathematical Monthly Vol 100, 1993, ISSN 0002-9890, pp. 331-340.
• Eric W. Weisstein: Taxicab Number. In: MathWorld (English).
• Taxicab number of Meyrignac in the Euler - network
• Taxicab numbers in the Euler - network
• Ivars Peterson about Taxicab numbers