A2 + B2

CELEBRATING 1980 with PYTHAGORAS

= C2


     The year 1980 is now upon us, so it is time to perform a little "mathemacation" on this number by way of recognizing its presence on our office calendars. Rather than celebrate in the usual manner of expressing all the integers from 1 to some higher value as an arithmetical combination on the digits, this paper illustrates what I hope is a novel approach. It takes the form of the following question:

"Of all the possible Pythagorean triples,
how many times does 1980 appear as one of their members?"

     Restated this asks: "In the Pythagorean equation A2 + B2 = C2, how many times does A or C equal 1980?" Of course, the reader is to understand that (a) B = 1980 is not considered as distinct from A = 1980; and (b) we are only interested in triples consisting of whole numbers.

     In light of (a) above, there are just two classes of answers, those for 19802 + B2 = C2 and those for A2 + B2 = 19802. The chart at the end shows that there are 67 triples for the first equation, but only one triple for the other.

     Instead of giving a detailed explanation of how the triples were obtained -- it is fairly straightforward using the three equations

A = 2mn,      B = m2 - n2,      and      C = m2 + n2

and a hand-held calculator -- I merely wish to point out some interesting number trivia that caught my eye during the whole process. The reader is invited to find other strange patterns and relationships of his own, as the discussion to follow is not intended to be exhaustive.

     1) The first, and perhaps the most unique, observation that should be made concerns the four digits in the number 1980 itself. Six triples are formed by using only those digits:

		No.  2:   1980	      189	  1989
		No. 38:   1980	     9801	  9999
		No. 39:   1980	    10800	 10880
		No. 59:   1980	    89089	 89111
		No. 61:   1980	   108891	108909
		No. 67:   1980	   980099	980101
Triple No. 38 has the added distinction that C2 (=99980001) is also composed of the same four digits. And the primitive form of No. 59 behaves similarly, namely (180, 8099, 8101).

     2) There are eight palindromes. Two of them (5775 and 7557) exhibit an obvious reversal of their digits. Two others (777 and 9999) are each composed of a single digit. The four remaining are 363, 1991, 2332 and 3003.

     3) Several terms exhibit permutations on sets of four distinct digits.

		1-4-5-8:  1584    (No. 1);	1485   (No. 12)
			  4581   (No. 25);	5148   (No. 27)
		2-4-5-7:  2475   (No. 12);	4752   (No. 27)
		1-2-7-8:  1728   (No. 14);	2871   (No. 16)
		1-2-5-7:  1275   (No. 11);	7125   (No. 32)
		3-5-7-9:  3597   (No. 21);	7395   (No. 32)
Of special note here is that Nos. 12 and 27 are closely interwoven with each other; that is, permuting the digits for the B and C values of one triple yields the C and B values, respectively, of the other. (A perfectly irrelevant sidelight to the first set is that 722 = 5184.)

     4) A more restricted variation on the above theme occurs in Nos. 28 and 52. Here the digits of the B and C terms are slightly switched around within the triple itself.

		No. 28:	B =  5265 and C =  5625
		No. 52:	B = 36273 and C = 36327
In fact, in No. 28, B is the exact reversal of C.

     5) The occurrence of consecutive digits shows up as follows:

		0-1-2-3:	2031  (No. 4)
		2-3-4-5:	4235 (No. 26)
		4-5-6-7:	4675 (No. 26)
		6-7-8-9:	8967 (No. 36)
In No. 18 we have the consecutive even digits of 0-2-4-6. And in the third category above we had consecutive odd digits.

     6) Other digital patterns of interest are the following:

     (a) "aabb": 1188 (No. 1); 2244 (No.9); 3300 (No. 18); 8800 (No. 35);
     (b) "abab": 7979 (No. 34); 9797 (primitive of B in No. 55);
     (c) "three-of-a-kind/in-a-row":

		 2225  (No. 8) 	 22231 (No. 48)	  5888 (No. 30)
		89111 (No. 59)	 98000 (No. 60)
     7) By "grouping" the digits without altering their given order, more clever effects can be seen. First, we can "see" three squares in these:

		 14916 (No. 42) gives    1  49    16
		 49025 (No. 55) gives   49   0    25
		196025 (No. 63) gives  196   0    25
The C term of No. 40 produces two squares, namely 121 and 81.

     Next, the insertion of operation signs and an equals sign leads us to these equations:

		  1 = 20 - 19,    from 12019 (No. 40)
		  4 = 89 - 85,    from 48985 (No. 55)
		 27 = 18 + 9,     from 27189 (No. 49)
		 27 = 26 + 1,     from 27261 (No. 49)
	 	 81 + 6 = 87,     from 81687 (No. 58)
		 12 = 25 - 13,    from 122513 (primitive of C in No. 66)
		  6  2 = 12,     from 6212  (No. 30)
		 24 + 5 + 0 = 29, from 245029 (No. 64)
     8) Finally, we present a few miscellaneous patterns dealing with squares and cubes.
		(a) in No. 14: 1728 = 123; and 12 is the number of
		    months in 1980.

		(b) in No. 37; 99012 = 98029801; this square is
		    formed from two consecutive integers, 9802 and 9801.

		(c) the primitive form of No. 46 (495, 4888, 4913)
		    yields 4952 + 48882 = 49132 = 24137569, a square
		    with eight distinct digits whose only (nonzero)
		    missing digit is 8; and 4913 = 173 with
		    4 + 9 + 1 + 3 = 17.

[from The Oregon Mathematics Teacher, January 1980, pp.12-13, 19]
===================================================================
PYTHAGOREAN TRIPLES WITH 1980
------------------------------------------------------------------- No. A B C No. A B C 1 1188 1584 1980 35 1980 8800 9020 2 1980 189 1989 36 1980 8967 9183 3 1980 209 1991 37 1980 9701 9901(p) 4 1980 363 2013 38 1980 9801 9999 5 1980 624 2076 39 1980 10800 10980 6 1980 777 2127 40 1980 12019 12181(p) 7 1980 825 2145 41 1980 12993 13143 8 1980 1015 2225 42 1980 14784 14916 9 1980 1056 2244 43 1980 16275 16395 10 1980 1232 2332 44 1980 17765 17875 11 1980 1275 2355 45 1980 18096 18204 12 1980 1485 2475 46 1980 19552 19652 13 1980 1541 2509(p) 47 1980 21735 21825 14 1980 1728 2628 48 1980 22231 22319 15 1980 2015 2825 49 1980 27189 27261 16 1980 2079 2871 50 1980 29667 29773 17 1980 2337 3063 51 1980 32640 32700 18 1980 2640 3300 52 1980 36273 36327 19 1980 2701 3349(p) 53 1980 39179 39229(p) 20 1980 2967 3567 54 1980 44528 44572 21 1980 3003 3597 55 1980 48985 49025 22 1980 3289 3839 56 1980 54432 54468 23 1980 3360 3900 57 1980 65325 65355 24 1980 3808 4292 58 1980 81663 81687 25 1980 4131 4581 59 1980 89089 89111 26 1980 4235 4675 60 1980 98000 98020 27 1980 4752 5148 61 1980 108891 108909 28 1980 5265 5625 62 1980 163344 163356 29 1980 5775 6105 63 1980 196015 196025 30 1980 5888 6212 64 1980 245021 245029(p) 31 1980 6384 6684 65 1980 326697 326703 32 1980 7125 7395 66 1980 490048 490052 33 1980 7293 7557 67 1980 980099 980101(p) 34 1980 7979 8221(p) 68 1980 400 2020 (p) denotes that the triple is "primitive" ==================================================================
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