Pythagorean Progressions (a^2+b^2+c^2...+m^2=n^2)

Pythagorean triples provide integer solutions to the equation:

a^2+b^2=c^2

Consider the equations:

a^2+b^2+c^2...+m^2=n^2

Propositions:

1) all such equations have multiple, nonrepetitive (i.e., a<>b<>c...<>m) solutions such that a,b,c...m,n are integer

Perhaps this is obvious enough. Consider rewriting the equations such that:

a^2+b^2+c^2... = n^2-m^2

Then, it can be noted that the sum on the left hand side of the equation (a^2, b^2, c^2,...) can formulated such that it produces an odd number (e.g., if a is odd and b & c are even, etc.).

The right hand side, n^2-m^2 can be formulated such that n=m+1. Since the difference of consecutive squares forms consecutive odd numbers, e.g.,:

1^2-0^2=1
2^2-1^2=3
3^2-2^2=5
4^2-3^2=7
etc.

it should be hopefully obvious that there will always be a rather large number of integer solutions for a, b, c, m, n.

Perhaps of more interest is the possibility that series can be formed that would appear to also satisfy this proposition. The tables below demonstrate several series that can satisfy the above proposition (for n=5 or greater). Beginning with row 2 in table 1 and row 4 in table 2, the m-1 term (shaded in light yellow) is formed by adding the previous two terms (e.g., for row 2, the m-1 term is 2+4=6; for row 3, 4+6=10, for row 4, 6+10=16, etc.). Term m (shaded in light red), for table 1 equals (((m*(m+(term m-1))-4)/2 (e.g., for row 2 (6*(6+4))-4)/2=28) (special thanks to David M. Einstein for demonstrating this). Term m for table 2 equals (((m*((m+(term m-1))+4)/2 (e.g., for row 4 (24*(24+14))+4)/2=458). The actual ratio of term m's from one row to the next is shaded in light green. Noting that the term m's are related to Fibonacci numbers (again, thanks to David for pointing this out), it is perhaps not surprising to note that this ratio approaches the square of the Golden Ratio (i.e., 2.618034). More examples of this type can be found in Pythagorean / Fibonacci Progressions ( http://mysite.verizon.net/reriker/madpythag2.html ).

TABLE 1

a b c d e f g h i j k l m n o p q r s t u
1 1 2 4 10 11
2 1 2 4 6 28 29 2.8
3 1 2 4 6 10 78 79 2.78571428571429
4 1 2 4 6 10 16 206 207 2.64102564102564
5 1 2 4 6 10 16 26 544 545 2.64077669902913
6 1 2 4 6 10 16 26 42 1426 1427 2.62132352941176
7 1 2 4 6 10 16 26 42 68 3738 3739 2.62131837307153
8 1 2 4 6 10 16 26 42 68 110 9788 9789 2.61851257356875
9 1 2 4 6 10 16 26 42 68 110 178 25630 25631 2.61851246424193
10 1 2 4 6 10 16 26 42 68 110 178 288 67102 67103 2.61810378462739
11 1 2 4 6 10 16 26 42 68 110 178 288 466 175680 175681 2.61810378230157
12 1 2 4 6 10 16 26 42 68 110 178 288 466 754 459938 459939 2.6180441712204
13 1 2 4 6 10 16 26 42 68 110 178 288 466 754 1220 1204138 1204139 2.6180441711709
14 1 2 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3152476 3152477 2.61803547433932
15 1 2 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 8253294 8253295 2.61803547433827
16 1 2 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 21607406 21607407 2.61803420549419
17 1 2 4 6 10 16 26 42 68 110 178 288 466 754 1220 1974 3194 5168 8362 56568926 56568927 2.61803411293332


TABLE 2

a b c d e f g h i j k l m n o p q r s t u
1 2 4 5 22 23
2 2 4 5 10 72 73 3.27272727272727
3 2 4 5 10 14 170 171 2.36111111111111
4 2 4 5 10 14 24 458 459 2.69411764705882
5 2 4 5 10 14 24 38 1180 1181 2.5764192139738
6 2 4 5 10 14 24 38 62 3102 3103 2.62881355932203
7 2 4 5 10 14 24 38 62 100 8102 8103 2.61186331399097
8 2 4 5 10 14 24 38 62 100 162 21224 21225 2.61960009874105
9 2 4 5 10 14 24 38 62 100 162 262 55546 55547 2.6171315491896
10 2 4 5 10 14 24 38 62 100 162 262 424 145434 145435 2.61826234112267
11 2 4 5 10 14 24 38 62 100 162 262 424 686 380732 380733 2.61790227869687
12 2 4 5 10 14 24 38 62 100 162 262 424 686 1110 996782 996783 2.61806730193417
13 2 4 5 10 14 24 38 62 100 162 262 424 686 1110 1796 2609590 2609591 2.6180147715348
14 2 4 5 10 14 24 38 62 100 162 262 424 686 1110 1796 2906 6832008 6832009 2.6180388490146
15 2 4 5 10 14 24 38 62 100 162 262 424 686 1110 1796 2906 4702 17886410 17886411 2.61803118497519
16 2 4 5 10 14 24 38 62 100 162 262 424 686 1110 1796 2906 4702 7608 46827242 46827243 2.61803469785161
17 2 4 5 10 14 24 38 62 100 162 262 424 686 1110 1796 2906 4702 7608 12310 122595292 122595293 2.61803357968424


Another example of a series is shown in the tables below:

TABLE 3

a b c d e f g h i j k l m n o p q r s t u
1 2 6 8 25 27
2 2 6 8 12 61 63 2.44
3 2 6 8 12 16 125 127 2.04918032786885
4 2 6 8 12 16 20 225 227 1.8
5 2 6 8 12 16 20 24 369 371 1.64
6 2 6 8 12 16 20 24 28 565 567 1.53116531165312
7 2 6 8 12 16 20 24 28 32 821 823 1.45309734513274
8 2 6 8 12 16 20 24 28 32 36 1145 1147 1.39464068209501
9 2 6 8 12 16 20 24 28 32 36 40 1545 1547 1.34934497816594
10 2 6 8 12 16 20 24 28 32 36 40 44 2029 2031 1.31326860841424
11 2 6 8 12 16 20 24 28 32 36 40 44 48 2605 2607 1.2838836865451
12 2 6 8 12 16 20 24 28 32 36 40 44 48 52 3281 3283 1.2595009596929
13 2 6 8 12 16 20 24 28 32 36 40 44 48 52 56 4065 4067 1.23895153916489
14 2 6 8 12 16 20 24 28 32 36 40 44 48 52 56 60 4965 4967 1.22140221402214
15 2 6 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 5989 5991 1.20624370594159
16 2 6 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 7145 7147 1.19302053765236
17 2 6 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 8441 8443 1.1813855843247


TABLE 4

a b c d e f g h i j k l m n o p q r s t u
1 6 10 12 69 71
2 6 10 12 16 133 135 1.92753623188406
3 6 10 12 16 20 233 235 1.75187969924812
4 6 10 12 16 20 24 377 379 1.61802575107296
5 6 10 12 16 20 24 28 573 575 1.51989389920424
6 6 10 12 16 20 24 28 32 829 831 1.44677137870855
7 6 10 12 16 20 24 28 32 36 1153 1155 1.39083232810615
8 6 10 12 16 20 24 28 32 36 40 1553 1555 1.34692107545533
9 6 10 12 16 20 24 28 32 36 40 44 2037 2039 1.31165486155827
10 6 10 12 16 20 24 28 32 36 40 44 48 2613 2615 1.28276877761414
11 6 10 12 16 20 24 28 32 36 40 44 48 52 3289 3291 1.25870646766169
12 6 10 12 16 20 24 28 32 36 40 44 48 52 56 4073 4075 1.23837032532685
13 6 10 12 16 20 24 28 32 36 40 44 48 52 56 60 4973 4975 1.22096734593666
14 6 10 12 16 20 24 28 32 36 40 44 48 52 56 60 64 5997 5999 1.20591192439172
15 6 10 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 7153 7155 1.19276304819076
16 6 10 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 8449 8451 1.18118272053684
17 6 10 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 9893 9895 1.17090779973961


2) all squares of odd integers greater than 3 may be formed by the sum of nonrepetitive integer squares

3) more speculative, all squares of odd integers greater than 3 may be formed by 5 or less nonrepetitive integer squares

Obviously, not offering a whole lot of proof of the above, just some beginning examples:

TABLE 5

a^2+b^2=c^2 a b c
3 4 5
5 12 13
8 15 17
7 24 25
9 40 41
11 60 61
13 84 85
15 112 113
17 144 145
19 180 181
a^2+b^2+c^2=d^2 a b c d
2 3 6 7
1 4 8 9
2 5 14 15
1 6 18 19
4 5 20 21
2 7 26 27
5 6 30 31
4 7 32 33
3 8 36 37
2 9 42 43
5 8 44 45
4 9 48 49
1 10 50 51
3 10 54 55
2 11 62 63
7 10 74 75
9 10 90 91
3 14 102 103
10 11 110 111
2 15 114 115
8 13 116 117
4 15 120 121
1 16 128 129
3 16 132 133
a^2+b^2+c^2+d^2=e^2 a b c d e
1 2 4 10 11
2 4 5 22 23
2 4 7 34 35
2 3 8 38 39
2 5 8 46 47
1 2 10 52 53
2 3 10 56 57
1 4 10 58 59
2 5 10 64 65
1 6 10 68 69
4 5 10 70 71
2 7 10 76 77
2 3 12 78 79
5 6 10 80 81
4 7 10 82 83
3 8 10 86 87
2 9 10 92 93
5 8 10 94 95
4 9 10 98 99
1 2 14 100 101
2 3 14 104 105
1 4 14 106 107
4 5 14 118 119
2 4 15 122 123
2 7 14 124 125
3 10 12 126 127
1 2 16 130 131
2 3 16 134 135
2 3 20 206 207
a^2+b^2+c^2+d^2+e^2=f^2 a b c d e f
1 2 4 6 28 29
2 4 7 8 66 67
2 4 5 10 72 73
2 3 8 10 88 89
2 5 8 10 96 97
1 2 4 14 108 109