## Ceiling Squares and Discrete Semiprimes

A discrete semiprime is the product of two distinct prime numbers. A discrete semiprime has exactly four divisors. If m=ab is a semiprime, with a and b the prime factors of m, and ordered so that a<b, then m has exactly four divisors which are, in numerical order: 1, a, b, and ab=m. The integer m then factors in exactly two ways: 1*m and ab. Consider again s2, the least perfect square greater than m, i.e., m’s ceiling square. How does this ceiling square relate to the two distinct factorizations of a discrete semiprime?

Let us restrict our attention to odd discrete semiprimes, so that the simplest case of a prime times two is eliminated.

The equation m=[(m+1)/2]2-[(m-1)/2]2 is an expression of m as the difference between two squares that corresponds to the factorization m=1*m. Every positive integer has this as an expression of itself. The discrete semiprimes have a second such expression corresponding to the factorization m=ab, viz.: m=[(a+b)/2]2-[(b-a)/2]2.

The only positive integer values of m for which [(m+1)/2]2 equals the ceiling square of m are, not too surprisingly, the first two primes, m=2 and m=3. However, there may be (this is not yet proven by the author) infinite cases where [(a+b)/2]2 equals the ceiling square for a discrete semiprime m=ab. The author has observed this to be true so far for the following discrete semiprimes less than 1024: 15, 21, 35, 55, 65, 77, 91, 143, 187, 209, 221, 247, 299, 323, 391, 437, 493, 551, 667, 713, and 851.

``````Those numbers’ factorizations:

15 = 3*5
21 = 3  *7
35 =   5*7
55 =   5  *11
65 =   5     *13
77 =     7*11
91 =     7   *13
143 =       11*13
187 =       11   *17
209 =       11      *19
221 =          13*17
247 =          13   *19
299 =          13      *23
323 =             17*19
391 =             17   *23
437 =                19*23
493 =             17      *29
551 =                19   *29
667 =                   23*29
713 =                   23   *31
851 =                   23      *37``````