March 14th, 2015, 10:55 pm
Consider the non-negative natural numbers: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19......encode primes as 1, the rest as 0E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1....The number E0 is001.10101000101000101.... (ie the primes)E0 is irrational in every base due to a simple proof which shows that the binary pattern of the primes, E, is not periodic.Since the location of the (binary) point is arbitrary, if we locate the point one place to the right, we obtain E1 (=0011.0101....).So E1/E0 is the integer 2.Since E0 is irrational in every number base, we are led to the interesting aphorism:Every integer greater than one is the ratio of two irrational numbers. Each of these irrational numbers expresses the infinite pattern of the prime numbers.The Brainteaser is: Prove that E0 is transcendentalNote that C0 = 110.01010111010111010.... (ie the composites)and E0+C0 = 111.11111111111111111......All contributions acknowledged.Regards Grenville