July 15, 2017

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By David Joyner, Richard Kreminski, Joann Turisco

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Unfortunately, no one knows for sure what a good estimate for what N0 (t:) is! One can re-interpret the result as stating that, for x large, the probability an integer near x is prime is about I ln (x) • CHAPTER 1 . SOME ELEMENTARY NUMBER THEORY 40 In spite of the fact that there are infinitely many primes, the general problem of showing that there are infinitely many primes of a certain type is very hard. We give a couple of examples of this idea. 4. (twin primes conjecture} There are infinitely many primes p for which p and p + 2 are prime.

To solve the above problem, we must find an integer t which gives nonnegative values for x and y. In other words, we wish to find t such that x = 1445 + 18t 2 0 , y = -2312 - 29t 2 0 . It turns out that the only value of t which satisfies these inequalities is t = -8 0 , which makes x = 5 and y = 8 . 5 Euler's phi function Euler's totient- or cf>-function is needed for the description of the RSA cryp­ tosystem given later. 9. The number of integers 1S aS b which are relatively prime to b is denoted f/J (b) , called the Euler cf>-function or the Euler totient function.

Otherwise, replace a by then next largest integer which has not been crossed out. If this new a is greater than v'N then stop. 5. Go to step 3. This process must terminate after at most v'N steps. 41n fact, a "polynomial time" primality test has recently ( August 2002) been announced by M. Agrawal, and two of his students, N. Kayal and N. Saxena, in the Dept. of Tech. in Kanpur. This is a very important result. However, at the time of this writing, it has not yet been published. 44 CHAPTER 1 . 5.

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