The positive rational numbers are all the numbers formed as the ratio of two positive integers, reduced to lowest terms for uniqueness by removing factors common to both the numerator and denominator. Some people say that the product when you multiply all the positive rational numbers together is 1. Is this true? If not, what is the value of the product? Also, given two positive rational numbers q2 > q1 > 0, what is the product of all the rational numbers on the closed interval [q1, q2] when multiplied together? What about the product of all the rational numbers in the open interval (q1, q2)?
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The following is Bob Conger’s detailed solution.
If we are seeking a desert of size k, begin by calculating (k + 1)! Call this result K. K is divisible by every integer from 1 to k + 1. Then, K + 2 is divisible by 2; K + 3 is divisible by 3; K + 4 is divisible by 4, and K + (k + 1) is divisible by (k + 1). Thus, we have k sequential integers, none of which are primes.
- How big can such a desert be? No limit, it can be as large as you want, as the construction above works for any value of k.
- A starting integer for a desert of size k is [(k + 1)!] + 2. This is not the smallest starting point for a desert of size k. A smaller one would be the product of all the primes between 1 and k + 1 (inclusive), plus 2. (Proof very similar to the above.)
- Maximum number of non-overlapping k-size deserts: infinite. Any integer multiple of K plus 2 would start a desert. (Proof essentially the same as above). Since K > k + 1, these deserts would not overlap one another.
Solutions were also submitted by Andrea Altomani and Clive Keatinge