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
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