## 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"

==================================================================
```