## Monday, January 14, 2013

### Odd Semi-prime Property

Let N = pq be any odd semiprime. Let 's' be the ceiling of square root of 'N'.
If 's' is even then there exist odd 'l' and k = l + 2 such that (s^2 - k^2) < N < (s^2 - l^2) else
If 's' is odd then there exist even 'l' and k = l + 2 such that (s^2 - k^2) < N < (s^2 - l^2)

Let r = q + Δ,
where 'Δ' is very small +ve integer compared to 'p' & 'q'

When 2l + 4 < p then pr will be less than and nearer to (s^2 + l^2) and p(r + 1) will be greater than and nearer to  (s^2 + k^2)

When 2l + 4 > p then pr will be greater than and nearer to (s^2 + l^2) and depends upon 2p, p(r + 1) will be either greater or lesser than and nearer to  (s^2 + k^2)

## Tuesday, January 8, 2013

### Odd Composite Property

Let N = pq be any odd composite. Let k = 1, 2, 3, ...., p, ...., q, ... n. Let u = (N- 1)/2 & v = u +1. Let x = u^2(mod k) and y = v^2(mod k). Then | x - y | = 0 when k = p or k = q else | x - y | != 0. Let p(r -1) < u^2 < pr where r is some integer and let ps < v^2 < p(s + 1) where s is some integer then s - r = q - 1 or s - r = q - 3.
Example 161 = 7*23, here u (161 -1)/2 = 80, v = 81, 80^2 = 2(mod 7) = 6(mod 23), 81^2 = 2(mod 7) = 6(mod 23). 7*914 < 80^2 < 7*915 and  7*937 < 81^2 < 7*938, 937 - 915 = 22 = 23 -1.