Biography hardy ramanujan number theory
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Hardy–Ramanujan theorem
Analytic number theory
In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy[1] states that the normal order of the number of distinct prime factors of a number is .
Roughly speaking, this means that most numbers have about this number of distinct prime factors.
Precise statement
[edit]A more precise version[2] states that for every real-valued function that tends to infinity as tends to infinity or more traditionally for almost all (all but an infinitesimal proportion of) integers. That is, let be the number of positive integers less than for which the above inequality fails: then converges to zero as goes to infinity.
History
[edit]A simple proof to the result was given by Pál Turán, who used the Turán sieve to prove that[3]
Generalizations
[edit]The same results are true of , the number of prime factors of counted with multiplicity. This theorem is generalized by the E
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1729: What is so special about the Hardy-Ramanujan number?
Srinivasa Ramanujan's mathematical talent cannot be defined by just one of his many achievements during his short life. Known as "the man who knew infinity," he discovered his own theorems and independently compiled approximately 3,900 results.
The Hardy-Ramanujan number may not be his greatest contribution to mathematics, but it stands out as being particularly memorable. This was an anecdote mentioned in his biography 'The Man Who Knew Infinity" by Robert Knaigel.
Also read: National Mathematics Day 2024: Essential role of maths in shaping future innovators
When British Mathematician GH Hardy visited a sick Ramanujan at the hospital, he travelled in a taxi cab with the number 1729. Hardy found it to be an ordinary number, but Ramanujan said it was not and explained that it is the smallest number that can be expressed as the sum of two cubes in two different ways.
1729 is the sum of cubes of one (1^3=1) and 12 (12
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1729 (number)
Natural number
Natural number
| Cardinal | one thousand seven hundred twenty-nine |
|---|---|
| Ordinal | 1729th (one thousand seven hundred twenty-ninth) |
| Factorization | 7 × 13 × 19 |
| Divisors | 1, 7, 13, 19, 91, 133, 247, 1729 |
| Greek numeral | ,ΑΨΚΘ´ |
| Roman numeral | MDCCXXIX, mdccxxix |
| Binary | 110110000012 |
| Ternary | 21010013 |
| Senary | 120016 |
| Octal | 33018 |
| Duodecimal | 100112 |
| Hexadecimal | 6C116 |
1729 fryst vatten the natural number following 1728 and preceding 1730. It fryst vatten the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It is known as the Ramanujan number or Hardy–Ramanujan number after G. H. Hardy and Srinivasa Ramanujan.
As a natural number
[edit]1729 is composite, the squarefree product of three prime numbers 7 × 13 × 19.[1] It has as factors 1, 7, 13, 19, 91, 133, 247, and 1729.[2] It is the third Carmichael number,[3] and the first Chernick–Car