Transcript A prime number (or a prime) is a natural number

PRIME
NUMBERS

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.
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.
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.
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 Rn,
Where n is serial number of that
Ramanujan prime.
Ramanujan prime (Rn ) satisfy following
result proved by ramanujan:
π(x) - π(x/2) ≥ n ,where n is serial number
of Rn.. for all x ≥Rn
HERE π(x) IS PRIME COUNTING
FUNCTION
π(x) shows the number of primes less
than or equal to x.
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
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.
List of Ramanujan Primes..
SOME EXAMPLES
R1= 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
 R2 =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
 R3 = 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
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
 10 = 3 + 7 = 5 + 5
 14 = 7 + 7 = 3 + 11
 26 = 13 + 13 = 7 + 19=3+23
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
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.
WEBSITES
 www.google.com
 www.wikipedia.org
 www.mathworld.wolfram.com
THANK YOU
BY:
PRIYANK KHETWANI
CHIRANJEEV DUA
KIRTI GOGIA
DASHLEEN KAUR