{"id":24109,"date":"2013-09-29T21:27:47","date_gmt":"2013-09-30T01:27:47","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=24109"},"modified":"2014-06-23T15:15:20","modified_gmt":"2014-06-23T20:15:20","slug":"legendres-conjecture","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/09\/29\/legendres-conjecture\/","title":{"rendered":"Legendre’s Conjecture"},"content":{"rendered":"
Adrien-Marie Legendre <\/a>(1752-1833), known for important concepts such as the Legendre polynomials and Legendre transformation, states that given an integer n > 0, there exists a prime number, p, between \"n^2\" and \"(n+1)^2\", or alternatively, \"n^2.\u00a0 This conjecture constitutes one of Edmund Landau’s
\nfour “unattackable” problems<\/a> mentioned at the 1912 Fifth Congress of Mathematicians in Cambridge.<\/div>\n
\n

A table of the results of the primes
\nthat exist between \"n^2\" and \"(n+1)^2\" can be seen below.<\/p>\n

<\/p>\n<\/div>\n

<\/div>\n
\n
\n\n\n\n\n\n\n\n\n\n\n\n
\n
\n
n<\/div>\n<\/div>\n<\/td>\n
\n
\n
Condition: \"n^2<\/div>\n<\/div>\n<\/td>\n
\n
\n
Prime
\nnumbers, p<\/div>\n<\/div>\n<\/td>\n
\n
\n
Number of primes between \"n^2\" and
\n\"(n+1)^2\"<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
1<\/div>\n<\/div>\n<\/td>\n
\n
\n
1
\n< p < 4<\/div>\n<\/div>\n<\/td>\n
\n
\n
2, 3<\/div>\n<\/div>\n<\/td>\n
\n
\n
2<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
2<\/div>\n<\/div>\n<\/td>\n
\n
\n
4
\n< p < 9<\/div>\n<\/div>\n<\/td>\n
\n
\n
5, 7<\/div>\n<\/div>\n<\/td>\n
\n
\n
2<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
3<\/div>\n<\/div>\n<\/td>\n
\n
\n
9
\n< p < 16<\/div>\n<\/div>\n<\/td>\n
\n
\n
11, 13<\/div>\n<\/div>\n<\/td>\n
\n
\n
2<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
4<\/div>\n<\/div>\n<\/td>\n
\n
\n
16
\n< p < 25<\/div>\n<\/div>\n<\/td>\n
\n
\n
17, 19, 23<\/div>\n<\/div>\n<\/td>\n
\n
\n
3<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
5<\/div>\n<\/div>\n<\/td>\n
\n
\n
25
\n< p < 36<\/div>\n<\/div>\n<\/td>\n
\n
\n
29, 31<\/div>\n<\/div>\n<\/td>\n
\n
\n
2<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
6<\/div>\n<\/div>\n<\/td>\n
\n
\n
36
\n< p < 49<\/div>\n<\/div>\n<\/td>\n
\n
\n
37, 41, 43, 47<\/div>\n<\/div>\n<\/td>\n
\n
\n
4<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
7<\/div>\n<\/div>\n<\/td>\n
\n
\n
49
\n< p < 64<\/div>\n<\/div>\n<\/td>\n
\n
\n
53, 59, 61<\/div>\n<\/div>\n<\/td>\n
\n
\n
3<\/div>\n<\/div>\n<\/td>\n<\/tr>\n
\n
\n
8<\/div>\n<\/div>\n<\/td>\n
\n
\n
64
\n< p < 81<\/div>\n<\/div>\n<\/td>\n
\n
\n
67, 71, 73, 79<\/div>\n<\/div>\n<\/td>\n
\n
\n
4<\/div>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
\n
We can see from the above table that indeed, the Legendre Conjecture holds true for the cases considered, with at least two prime numbers, p, fulfilling the inequality, \"n^2.\u00a0 Furthermore, the positive integers, \"n^2\" and \"(n+1)^2\" are referred to as square numbers.\u00a0 But, how would one go about proving the conjecture true for any n?The best effort so far, it seems, was given by Chinese mathematician, Chen\u00a0Jingrun\u00a0(\u9673\u666f\u6f64) (1933-1996),\u00a0who proved a slightly weaker version of Legendre’s Conjecture: there is either a prime, p in the interval, \"n^2 or a\u00a0semiprime, pq, in the interval, \"n^2, where q is a prime not equal to p. \u00a0A semiprime is a composite number > 1 (not prime) that is the product of two prime numbers (could even possibly be equal primes). Examples of the initial semiprime numbers are 4, 6, 9, 10, 14, 15, 21, 22, etc. Examples of the first several semiprime numbers whose factors are distinct and for which Jingrun proved a slightly weaker version of Legendre’s Conjecture are: 6, 10, 14, 15, 21, 22, 26, 33, 34, etc. If the Legendre Conjecture<\/a> turns out to be true, then the gap between any two primes, \"p_n\" and \"p_{n+1}\" would be \"O(p^{1\/2})\", where p is a prime number, and O constitutes the big O
\nnotation.\u00a0\u00a0Understanding\u00a0the big O notation, let’s look at the table above for the case n\u00a0= 2, for which it is found that the prime numbers 5 and 7 fall within the interval, \"n^2; therefore the gap between 5 and 7 should be \"5^{1\/2}\" \u00a0which is about 2.236.\u00a0 Accordingly, the gap between prime numbers in big O notation has to be adjusted to become more representative of the\u00a0actual gap between prime numbers if the Legendre Conjecture is true.This problem definitely seems hard to tackle,
\nbut time will tell whether it will be proven, whether rigorously in analytic sense or with the aid of computers in a numerical fashion.What are your thoughts on approaching Legendre’s Conjecture?<\/p>\n<\/div>\n<\/div>\n
<\/div>","protected":false},"excerpt":{"rendered":"

Adrien-Marie Legendre (1752-1833), known for important concepts such as the Legendre polynomials and Legendre transformation, states that given an integer n > 0, there exists a prime number, p, between and , or alternatively, .\u00a0 This conjecture constitutes one of … Continue reading →<\/span><\/a><\/p>\n

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