PRIME NUMBERS : The Most Mysterious Figures in Math 

Prime numbers have always fascinated mathematicians. They appear among the integers seemingly at random, and yet not quite: there seems to be some order or pattern, just a little below the surface, just a little out of reach.
—Underwood Dudley

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 11 is a prime number because it has no positive divisors other than 1 and 11.

Lets Start

Properties and Facts 

1)There are infinite prime numbers.

2) There are infinite number of twin primes(The set of N,N+2 such that both of them are primes)

3) There is exactly one prime triplet N, N+2, N+4 where N = 3.

4) A prime number p, p > 3, is always of the form 6k ± 1 .

5) 1-10 : 4 primes ; 1-100 : 25 primes ; 1-10³ : 168 primes ; 1- 10⁴ : 1229 primes.[Trivia. Sometimes useful.]

6) If A,B are primes then either A – B or A + B is divisible by 3. Also, if A,B,A – B,A + B are all primes then there is only one solution, i.e. A = 5,B = 2

7) The first 25 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

8) A Fermat Prime is a prime of the form 2ⁿ +1 .

9) A Mersenne Prime is a prime of the form 2ⁿ -1 .

10) As of October 2018, the largest known prime number is 2^77,232,917 − 1, a number with 23,249,425 digits. It was found in December 2017 by the Great Internet Mersenne Prime Search (GIMPS) .

Procedure to find out the prime number :

Suppose A is a given number.

Step 1: Find a whole number nearly greater than the Square root of A.

Step 2: Check whether A is divisible by any prime number less than square root of A .

If yes, A is not a prime number. If not, A is prime number.

Problems for practice :

1. Exactly one of the five numbers listed below is a prime number. Which one is the prime number?
(A) 999,991
(B) 999,973
(C) 999,983
(D) 1,000,001
(E) 7,999,973

2. How many prime numbers p are there such that 2p+1 is the square of an integer?

3. For how many prime numbers p, is p² + 3p -1 a prime number?

4. The number 104,060,465 is divisible by a five-digit prime number. What is that prime number ?

5. Suppose there exists a positive integer N such that there are 2016 prime numbers that are less than N.

(i) Find the remainder when the sum of the 2016th powers of all primes less than N is divided by 6.
(ii)Find the remainder when the sum of the 2016th powers of all primes less than N is divided by 24.

6. For how many positive integers ‘n’, is [n²/3]  a prime number, where [x] is the greatest integer less than or equal to x?

7. A three digit number ABC is divisible by 11 and all its digits are distinct prime numbers. How many such three digit numbers exist?

8. Suppose that p and q are two-digit prime numbers such that p² – q² = 2p + 6q + 8. Find the largest possible value of p + q. 

9. Find the only prime number that can divide two successive integers of the form n² + 3 ?

10. Find the largest integer for which 2n² – 29n+77 is a prime number. 

11. A number N has 3 prime factors (2, 3, 5), and 27 factors which are perfect cubes. If 64 of the factors of N are perfect squares, how many different values can N take ?

(A) 1
(B) 2
(C) 4
(D) 8
(E) None of the above

12. What can be said about the number 9999999 + 1999000 ?

(i) Its is prime
(ii) It is composite 

13. How many of the 3 numbers
(i)2²⁴⁵+1
(ii)2²⁴⁶+1
(iii)2²⁴⁷+1
are prime?

14. How many prime factors does the number 2 + 2² + 2³ + 2⁴ +……+ 2¹⁶ have ?

15. Find all the primes of the form n³+ 1 .

16. Number N has 3 prime factors, and 27 factors which are perfect cubes. If 125 of the factors of N are perfect squares, how many factors does N have ?

(A) 648
(B) 729
(C) 900
(D) 1000
(E) None of these 

17. Find all prime numbers p such that 7p + 4 is a perfect square and justify your answer.

18. N is the product of the first 50 prime numbers. A is a factor of N and B is a factor of A. How many ordered pairs (A, B) exist?

19. If p and q are two prime numbers and p × q = 1001 × 1003 – 120, find the value of p + q.

20. For how many positive integers ‘n’ each of the numbers n + 1, n + 3, n + 7, n + 9, n + 13 and n + 15 is prime? 

[ Post your answers in the comment section ]

Need more dope? :
Divisibility Rule of a Number (Divisibility Test)
Nifty Formulae for Natural Numbers
Divisors of a Number ( Number of Divisors of a number)

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