2. PRIME NUMBERS

3.  FOR EXAMPLE 7 IS A PRIME NUMBER BECAUSE ONLY 1 AND 7 EVENLY DIVIDE IT.  NUMBERS WHICH HAVE MORE FACTORS OTHER THAN 1 AND THE NUMBER ARE COMPOSITE NUMBERS.  1 IS NEITHER PRIME NOR COMPOSITE BECAUSE 1 HAS ONLY ONE FACTOR THAT IS 1 ITSELF.

5. A BRIEF… Srinivasa Ramanujan (Tamil: ஸ்ரீனிவாஸ ராமானுஜன் ; 22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. He stated results that were both original and highly unconventional, such as the RAMANUJAN PRIME and the RAMANUJAN THETA FUNCTION, and these have inspired a vast amount of further research.

8. TO UNDERSTAND RAMANUJAN PRIME WE NEED TO UNDERSTAND FOLLOWING: The prime counting function shows number of primes less than or equal to x. It is denoted by π(X). Examples π(2)= 1 which is only 2 π(3)= 2 which are 2 & 3 π(7)= 4 which are 2,3,5 & 7.

9. RAMANUJAN PRIME In Mathematics, a Ramanujan prime is a prime number that satisfies a result proven by SRINIVASA RAMANUJAN. It relates to the prime counting function. And Ramanujan prime is denoted by R n, Where n is serial number of that Ramanujan prime.prime counting function

10. Ramanujan prime (R n ) satisfy following result proved by ramanujan: π(x) - π(x/2) ≥ n,where n is serial number of R n.. for all x ≥R n HERE π(x) IS PRIME COUNTING FUNCTION π(x) shows the number of primes less than or equal to x.

11. In actual RAMANUJAN PRIME is modification of BERTRAND & CHEBYSHEV THEOREM which was: There is at least one prime p such that x < p <2x, if x ≥2

12. SIMPLY WE CAN SAY….. That with help of Ramanujan prime we can find minimum number of prime numbers that exist between a number & its half.

13. List of Ramanujan Primes..

15. SOME EXAMPLES R 1 = 2 NOW WE CAN TAKE ANY X ≥ 2 SAY X= 3 π(x) - π(x/2) ≥ n π(3) - π(3/2) ≥ 1 2-0 ≥ 1 2 ≥ 1

16.  R 2 =11NOW WE CAN TAKE ANY X ≥ 11 SAY X= 13 π(x) - π(x/2) ≥ n π(13) - π(13/2) ≥ 2 6-3 ≥ 2 3≥2

17.  R 3 = 17 NOW WE CAN TAKE ANY X ≥ 17 SAY X= 20 π(X) - π(x/2) ≥ n π(20) - π(20/2) ≥ 3 8-4 ≥ 3 4 ≥3

18. Goldbach’s Conjecture  The conjecture states: Every even integer greater than ‘2’ can be expressed as the sum of two primes.  Let us take a number 2n where ‘n’ be any integral value other than 1, then the number can be expressed as sum of ‘p’ and ’q’, both of which are primes, i.e., 2n = p + q For eg - 8 = 3 + 5 8 = 3 + 5 10 = 3 + 7 = 5 + 5 10 = 3 + 7 = 5 + 5 14 = 7 + 7 = 3 + 11 14 = 7 + 7 = 3 + 11 26 = 13 + 13 = 7 + 19=3+23 26 = 13 + 13 = 7 + 19=3+23

19. Ramanujan on studying the conjecture concluded the folloiwng result EVERY EVEN INTEGER GREATER THAN 2 CAN BE EXPRESSED AS THE SUM OF ATMOST FOUR PRIME NUMBERS. Examples……. 100 = 5 + 11 + 23 + 61 170 = 19 + 23 + 61 + 67 804 = 127 + 157 + 257 + 263 And the list goes on for all even integers > 2

20. The unique findings of Ramanujan: He found out the smallest and the largest number which could be expressed as-  A product of three primes  Sum of cubes of two numbers with two distinct pairs of numbers. Smallest Number 1729 can be expressed as:  7 x 1 3 x 19  1 3 + 12 3 or 9 3 + 10 3 Largest Number 885623890831 can be expressed as:  3943 x 14737 x 15241  7511 3 + 7730 3 or 8759 3 + 5971 3

21. WEBSITES  www.google.com www.google.com  www.wikipedia.org www.wikipedia.org  www.mathworld.wolfram.com www.mathworld.wolfram.com

22. THANK YOU BY: PRIYANK KHETWANI CHIRANJEEV DUA KIRTI GOGIA DASHLEEN KAUR