## Integer Factorization: Look at the Data

I am returning my attention, amid dealing with holiday decoration and winter coldness precaution, to the numbers both small and largish, in examining discrete semiprimes for patterns that may give clues to factorization. The table/scribble-sheet below, completed without automated computation beyond a calculator, is my current view of the behaviors of the smallest ones that do not immediately give Fermat factorization based on their Ceiling Squares. This is, I realize, looking at the easily known to try to figure out something new about the inscrutably unknown.

``````Discrete Semiprime Studies
Starting with the Smallest of Them
Which Do Not Break Out Instantly
Via Fermat Factorization
Based on Their Adjusted Ceiling Squares
(prime numbers up to 41 as factors)

ASCII graphics like "X...*..X" show remainder distances between perfect squares.
"Tr:" refers to the trivial factorization, with diff from non-trivial above it.

51 = 8^2-13	X...*..X
= 10^2-7^2.
Tr:  26^2-25^2 (+16,+18)

57 = 9^2-24	X.......*X
= 11^2-8^2.
Tr:  29^2-28^2 (+18,+20)
95 = 10^2-5	X*...X
= 12^2-7^2.
Tr:  48^2-47^2 (+36,+40)

69 = 9^2-12	    X..*...X
= 11^2-52	X..*...........X
= 13^2-10^2.
Tr:  35^2-34^2 (+22,+24)
115 = 12^2-29	X...*......X
= 14^2-9^2.
Tr:   58^2-57^2 (+44,+48)
161 = 13^2-8	X...*X
= 15^2-8^2.
Tr:   81^2-80^2 (+66,+72)

87 = 10^2-13	       X...*..X
= 12^2-57	   X.......*......X
= 14^2-109	X........*...........X
= 16^2-13^2.
Tr:  44^2-43^2 (+28,+30)
145 = 13^2-24	    X.......*X
= 15^2-80	X...............*X
= 17^2-12^2.
Tr:   73^2-72^2 (+56,+60)
203 = 16^2-53	X...*..........X
= 18^2-11^2.
Tr:   102^2-101^2 (+84,+90)
319 = 18^2-5	X*...X
= 20^2-9^2.
Tr:   160^2-159^2 (+140,+150)

93 = 11^2-28	      X..*.......X
= 13^2-76	   X...........*....X
= 15^2-132	X..........*...........X
= 17^2-14^2.
Tr:  47^2-46^2 (+30,+32)
155 = 14^2-41	    X....*.......X
= 16^2-101	X*...................X
= 18^2-13^2.
Tr:   78^2-77^2 (+60,+64)
217 = 15^2-8	      X...*X
= 17^2-72	X.......*........X
= 19^2-12^2.
Tr:   109^2-108^2 (+90,+96)
341 = 19^2-20	X...*....X
= 21^2-10^2.
Tr:   171^2-170^2 (+150,+160)

111 = 12^2-33	         X.......*..X
= 14^2-85	     X...*..............X
= 16^2-145	  X*.......................X
= 18^2-213	X................*...........X
= 20^2-17^2.
Tr:   56^2-55^2 (+36,+38)
185 = 15^2-40	      X...*........X
= 17^2-104	   X..*................X
= 19^2-176  X......*...................X
= 21^2-16^2.
Tr:   93^2-92^2 (+72,+76)
259 = 18^2-65	   X*...............X
= 20^2-141	X...................*..X
= 22^2-15^2.
Tr:   130^2-129^2 (+108,+114)
407 = 22^2-77	X............*...X
= 24^2-13^2.
Tr:   204^2-203^2 (+180,+190)
481 = 23^2-48	X...........*X
= 25^2-12^2.
Tr:   241^2-240^2 (+216,+228)

123 = 12^2-21	            X....*...X
= 14^2-73           X........*.......X
= 16^2-133	     X...........*..........X
= 18^2-201    X....*.......................X
= 20^2-277	X....................*...........X
= 22^2-19^2.
Tr:   62^2-61^2 (+40,+42)
205 = 15^2-20	           X...*....X
= 17^2-84	      X..*...............X
= 19^2-156	   X...........*............X
= 21^2-236	X..........*...................X
= 23^2-18^2.
Tr:   103^2-102^2 (+80,+84)
287 = 18^2-37	        X*...........X
= 20^2-113	    X............*.......X
= 22^2-197	X*...........................X
= 24^2-17^2.
Tr:   144^2-143^2 (+120,+126)
451 = 22^2-33	      X.......*..X
= 24^2-125	X...*..................X
= 26^2-15^2.
Tr:   226^2-225^2 (+200,+210)
533 = 25^2-92	X..........*.......X
= 27^2-14^2.
Tr:   267^2-266^2 (+240,+252)
697 = 27^2-32	X......*...X
= 29^2-12^2.
Tr:   349^2-348^2 (+320,+336)
779 = 28^2-5	X*...X
= 30^2-11^2.
Tr:   390^2-389^2 (+360,+378)
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