On Prime-to-Prime Numbers

          On Prime-to-Prime Numbers

          Once, at the gym, I realized that I’d rather do 12 rounds of push-ups than 10; and this is because I would count off the rounds, and I liked it when the fraction of the total reduced. (The reductions took my mind off the push-ups. I like having done push-ups, but I hate doing them.)

2 rounds of 12 is one-sixth; 3 is one-fourth; 4 is one-third; 6 is one-half; 8 is two thirds; 9 is three quarters; and 10 is five sixths. But with 10 rounds, the only fractions that reduce are 2/10, 4/10, 5/10, 6/10, and 8/10. 3, 7, and 9 are relatively prime to 10, but 9 is composite; whereas 5, 7, and 11 are relatively prime to 12, and all of them are prime.

So of all the numbers between 1 and 12, twelve is relatively prime only to primes. This is a property not shared by ten; therefore my preference for 12 over 10 push-ups.

          So define a prime-to-prime number (or workout number) as a number W such that, if N is an integer between 1 and W, and N is relatively prime to W, then N is prime.

          Here are some theorems:

          Any prime-to-prime number is less than the square of the smallest prime not in its prime factorization.

          Proof:

          If W is prime-to-prime, and p is the smallest prime not in its factorization, then W is relatively prime to p², which is not a prime. So by definition of prime-to-prime, W is not greater than p²; and by definition of p, W does not equal p²; therefore W is less than p².

QED.

          The largest prime-to-prime number is 30.

          Proof:

          If W is prime-to-prime, and W>30, then W>4=2² ; and W>9=3² ; and W>25=5². Therefore W has factors 2, 3, and 5:

          W = 2*3*5*k  =  30k,   for some k.

          So if W>30, then W is at least 60, so W>49=7². Therefore W has factors 2, 3, 5, and 7:              W = 2*3*5*7k  =  210k,   for some k.

          So W is at least 210; so W>121=11². Therefore W has factors 2, 3, 5, 7, and 11:              W = 2*3*5*7*11k  =  2310k,   for some k.

          And so on! W has factors 2, 3, 5, 7, 11, endlessly!*

          Therefore W is infinite, that is, nonexistent.

          Therefore there are no prime-to-prime numbers greater than 30.

          QED.

          *Note: The product 2*3*5*…*p_n  (for p_n the nth prime) is less than (p_n+1)², for n>4. This is because:

          2*3*5*…*p_n      =       3*…*5p_n-1*2p_n

                                      >       3*…*4p_n-1*p_n+1

                                      >       3*…*2*p_n*p_n+1

                                       >       3*…*p_n+1*p_n+1

                                       >       (p_n+1)²

          The first three inequalities are due to Bertrand’s Theorem:

                                      2p_n-1  >  p_n

There are ten prime-to-prime numbers:

1, 2, 3, 4, 6, 8, 12, 18, 24, 30

          Proof is by inspection of all integers from 1 to 30.