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 (number)
Natural number
Natural number
| Cardinal | one thousand sju 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 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It fryst vatten known as the Ramanujan number or Hardy–Ramanujan number after G. H. Hardy and Srinivasa Ramanujan.
As a natural number
[edit]1729 fryst vatten 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
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Quick Info
Cranleigh, Surrey, England
Cambridge, England
Biography
G H Hardy's father, Isaac Hardy, was bursar and an art master at Cranleigh school. His mother Sophia had been a teacher at Lincoln Teacher's Training School. Both parents were highly intelligent with some mathematical skills but, coming from poor families, had not been able to have a university education. Hardy (he was always known as Hardy except to one or two close friends who called him Harold) attended Cranleigh school up to the age of twelve with great success [6]:-His parents knew he was prodigiously clever, and so did he. He came top of his class in all subjects. But, as a result of coming top of his class, he had to go in front of the school to recei