The product of two primes, P one and P two, then phi of N is just the value of phi forĮach prime multiplied together, or P one minus one, times P two minus one. That is, phi A times BĮquals phi A times phi B. Result based on the fact that the phi function Phi of 21,377, a prime number, you would only need to subtract one to get the solution, 21,376. To calculate phi of seven, a prime number, we count all integers, except seven, since none of them shareĪ factor with seven. Since prime numbers have noįactors greater than one, the phi of any prime number, P, is simply P minus one. Now, notice any predictable pattern? The straight line of points along the top represent all the prime numbers. It is a plot of values of phi, over integers from one to 1,000. What's interesting is that calculating the phi function is Notice six is not counted, because it shares a factor of two, while one, three, fiveĪnd seven are all counted, because they only share a factor of one. The Euler phi function satisfies the multiplicative property ( x y ) ( x ) ( y ) if the two integers x and y are relatively prime (also known as. To find the phi of eight we look at all values from one to eight, then we count how many integers eight does not share aįactor greater than one with. So, given a number, say N, it outputs how many integersĪre less than or equal to N that do not share anyĬommon factor with N. It measures the breakability of a number. One important function he defined is called the phi function. If we want to find phi(x) then we have to divide it by all its prime divisor then multiply the result with (that prime number-1).So. Continued to investigate properties of numbers, specifically the distribution of prime numbers.

Graph of euler totient function