If n is the product of 2 distinct primes then every integer k smaller than n has a rhythm. If n is any other integer, then all k < n such that GCD(k,n) = 1 have a rhythm. Of particular interest are the integers with the shortest or the smallest possible rhythm.

Since r must be greater than 1 by definition then the smallest possible rhythm r is 2, and therefore the integers with the smallest rhythm are the integers, for which k^2 = k mod n

**Claim:**If n is the product of two distinct primes p < q then (((n-1)/2)+1)

^{2}has exactly 4 square roots.

^{ }

**Claim:**If n is the product of two distinct primes p < q and m=(((n-1)/2)+1) + TheSmallestSquareRootOf((((n-1)/2)+1)

^{2}mod n) mod n then m is either equal to q or m is a small multiple of q such that m

^{2}= m mod n

Furthermore, if w = (((n-1)/2)+1) - TheSmallestSquareRootOf((((n-1)/2)+1)

^{2}mod n) mod n then w is either equal to p or w is a small multiple of p such that w

^{2}= w mod n

Example:

i) Let n = 35 so p = 5, q = 7

then m = (((35-1)/2)+1) + TheSmallestSquareRootOf((((35-1)/2)+1)

^{2}mod 35) mod 35

= (18 + TheSmallestSquareRootOf((18)

^{2}mod 35) mod 35

= (18 + TheSmallestSquareRootOf(9) mod 35) mod 35

= (18 + 3) mod 35

= 21 mod 35

and 21

^{2}= 21 mod 35, also 21 = 7*3

ii) Let n = 77, so p = 7, q = 11

then m = (((77-1)/2)+1) + TheSmallestSquareRootOf((((77-1)/2)+1)

^{2}mod 77) mod 77 =

= (39 + TheSmallestSquareRootOf((39)

^{2}mod 77) mod 77

= (39 + TheSmallestSquareRootOf(58) mod 77) mod 77

= (39 + 17) mod 35 [since 17

^{2}= 58 mod 77]

= 56 mod 35

and 56

^{2}= 56 mod 77, also 56 = 7*8

So an algorithm for finding an integer m with the smallest rhythm would involve the following steps:

- Calculate k=(((n-1)/2)+1)
- Calculate b = the smallest square root of((((n-1)/2)+1)
^{2}mod n) - Return m = k + b mod n