Presentation about number theory importance ppt.pptx

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1. Number Theory Rahim ud din Theorems and applications of prime numbers.

2. Important Theorems There are infinitely many primes Every integers greater then 1 has a prime factors. If n is a composite integer, then n has a prime factor not exceeding Suppose that a and b are relatively prime positive integers. Then the arithmetic progression an + b, n = l, 2, 3, ... , contains infinitely many primes. Prime number theorem, Twin Prime Conjecture There are infinitely many pairs of prim.es p and p +2.  

3. 7 . The Erdos Conjecture on Arithmetic Progreuions of Primes. For every positive integer n > 3, there is an arithmetic progression of primes of length n. 8. Goldbach's Conjecture. Every even positive integer greater than 2 can be written as the sum of two primes. 9. The n^2 + 1 Conjecture. There are infinitely many primes of the form n2+1, where n is a positive integer. 10. The Legendre Conjecture. There is a prime between every two pairs of consecutive squares of integers.

4. Applications of Prime Numbers in Real-Life Cryptography RSA Key Exchange Internet Security Credit Card Transactions Mathematical Research (Cryptography , Computer Science, and Engineering .)

5. Cryptography  you have a secret message that you want to send to your friend over the internet. You want to make sure that no one else can read or tamper with the message. This is where cryptography can help you secure it.

6. RSA ( Rivest-Shamir- Adleman ) Suppose we multiply two prime numbers together to make a big number. This big number will act as our secret code, which is known as the  Public Key . We keep one of the original prime numbers a secret (which is our private key). When our friend wants to send a secret message, they use our  Public Key  to scramble it. Only our private key can unscramble the message which makes our message secure . e.g. 0 and 1 are public key. 7152015208520118752011851 Go to the target.

7. Key Exchange Prime numbers also helps us  in secure and safe web browsing with the help of a process known as Key Exchange.  When we visit any secure website (like online banking site, government sites, etc ), our computer and the website need to agree on a secret code to keep the communication private and safe .

8. Internet Security Prime Numbers  are used in different fields to provide us  Internet security . Prime numbers  also play a role in creating digital signatures.  Have you ever wondered how the transactions you make online stay safe? Prime numbers are also very important in this case. They assist in encrypting your credit card data. When we make any credit card transaction, our digital information stored in the card is used to verify that the transaction has not been tampered which makes it nearly hard for fraudsters to intercept and exploit our personal data and information.

9. Games Prime Number Trick. 1. P ick any prime number greater than 3. 2. Square it . 3 . Add 14 . 4 . Divide by 12.

10. OBSERVATIONS: • All the three digit numbers on the left hand side are divisible by 37. • All the numbers on the right hand side are divisible by the number 3. • 111 / 37 = 3 • 222 / 37 = 6 • 333 / 37 = 9 • 444 / 37 = 12 • 555 / 37 = 15 • 666 / 37 = 18 • 777 / 37 = 21 • 888 / 37 = 24 • 999 / 37 = 27

11. Question and Answer Session