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The AMTI is a pioneer organisation in promoting and conducting Maths Talent Tests in India. Last year (43rd TC Data) (in the 43rd National level tests) 54058 students from 332 institutions spread all over India, participated at the screening level; 10% of them insitutionwise were selected for the final test. For the benefit of final level contestants and the chosen few for INMO, special orientation camps were conducted. Merit certificates and prizes were awarded to the deserving students.
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NMTC Question Paper
Let be a sequence of nondecreasing positive integers such that for some positive integers and . Prove that there exists a positive integer such that
In today’s post , I will present a very well known proof. What makes this proof especially appealing is that it is not too complex. Even so, it is very powerful. This theorem was first presented in Euclid’s Elements (Book IX, Proposition 20).
Theorem: There are an infinite number of primes.
(1) Assume that there is only a finite number of primes.
(2) Then, there exists a prime pn that is the largest prime.
(3) Let p1, p2, …, pn be the list of all primes that exist.
(4) Let x = p1*p2*…*pn + 1.
(5) By the fundamental theorem of arithmetic (see Theorem 3, here), we know there is at least one prime that divides x. Let us call this prime p*.
(6) But none of the primes p1 … pn divide x since x ≡ 1 (mod pi) for any of the primes p1 … pn
(7) Therefore, we have a contradiction. We have a prime p* that is not in the complete list of primes.
(8) So, we reject our assumption in step#1 and conclude that there are an infinite number of primes that exist.
The largest prime number till found in
243112609 – 1 ( mersenne primes )
This number was discovered by G1 year 2008 . This number has total of 12978189 digits . Landon Curt Noll computed the decimal value of this prime with calc and the English name with number.
Here is the top 10 mersenne primes known
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