submitted to the Journal Nature
Ancient Super-computing:
The Square Root of 2 Calculated to 40 sexagesimal digits in the sands of Sumeria?
In Memorium, Dr. Clay Perry, University of California, professor emeritus
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59:59:59:
by Richard Allen Brown, Dr. John McKee
1248 Insititue, Charleston, and Santa Barbara
Powered by Mathematica
Summary
The purpose of this piece is to show that that mathematicians of ancient Sumaria(circa 2000 BC.) could calculate the square root of 2 to great accuracy--the equivalent of 10 or more decimal digits. They could make these calculations using mathematics derived from their knowledge of ordered sequences of primitive Pythagorean triples. Additonally, a general formula will be developed using sequences of Pythagorean triples for finding the the square root of any prime number. Pythagorean triples are right triangles formed by integers only. Primitive Pythagorean triples are those triples with whose sides have no factor in common.
These results will be applied to two questions of historical mathematics. First,the calculation of the sexagesimal value 1:24:51:10: for the square root of 2 in the tablet number 7289 of the Yale Babylonian Collection. And Second,
the values used by Archimedes for the square root of 3 used in finding his estimate of pi. Archimedes' upper and lower bounds were 265/153 and 1351/780
for the square root of 3 given without explanation or derivation.
Finally, convergent sequences of Pythagorean triples are shown to be another possible early development of infinite processes in ancient mathematics.
How could the ancient Sumarians calculate the square root of 2 to great accuracy?
Table 1 shows the simple mathematical progression that leads to a ratio of two integers that estimate the square root of 2 to over seventy decimal digits of accuracy. Our sequence was actually calculated using the computer program Mathematica, although the calculations are so simple that they could be performed by grade school children.
For the ancient Sumarians to have made these calculations, they would have had to know the sequence of primitive Pythagorean triples which converged to a 45 degree triangle. The first few of these triples are (3,4,5), (20,21,29), (119,120,169), and (695,696,985). These triangles and many, many more were known by the ancient mathematicians of Sumeria. The Plimpton 322 tablet(circa 1900 BC) of the Columbia University collection indicates knowledge of at least the first few thousand primitive Pythagorean triples.
As proved in Euclid's Elements(circa 300 BC) there is associated with each
primitive Pythagorean triple a unique pair of integers, one odd and one even
which generate the triple by the formulas: a**2+b**2 ,a**2-b**2 ,and 2ab, where a and b are the two integers. Hence, if you know the sequence of pairs of generators that produce the primitive triples, the calculation can be made!
In fact, there is a very simple relationship for the sequence of generators as shown in table 1. The formula is k(n)=2*k(n-1)+k(n-2), where n is the sequence number and k(n) is the value of the nth generator. Each successive pair of generators produces a unique triple that more and more closely approximates a 45 degree triangle! The triangle being approximated is (1,1,2**(1/2)), this triangle cannot have three integer sides, but we have produced an
infinite sequece of integer triangles that produces an arbitrarily close result. How close? As our table shows, after over one hundred pairs of generators we have reached an estimate of the square root of 2 to 80 decimal places or over 40 sexagesimal places.
One additional mathematical step is needed to make the computation complete. While we could use the generators to produce the integer triangles and then calculate the estimate of the square root of 2 using the triangles, in fact, the generators themselves form an ever closer approximation to the ratio 1 + (2**(1/2) from which the square root of 2 can easily be derived. The reason this relationship is used is that it is general for all integers as will be shown later. The reader can verify that the generated triangles also converge rapidly to 45 degree triangle.
Did the ancient Sumerians know this computational method? We don't know how early this relationship was known, the authors have some evidence(shown below) that it may have been known or discovered by Archimedes(circa 300 BC). However, if the Sumarians had known the formula they could have performed the calculations to generate the sequence of numbers.
Table 1.
The first 104 generators of the convergent sequence of Pythagorean Triples to a 45 degree right triangle
| Sequence | Generators | (2**(1/2)+1)-1 Estimate |
|---|
| n | k(n)=2k(n-1)+k(n-2) | (k(n)/k(n-1))-1 |
|---|
| 0 | 1 |
|---|
| 1 | 2 | 1 |
|---|
| 2 | 5 | 1.5 |
|---|
| 3 | 12 | 1.4 |
|---|
| 4 | 29 | 1.41 |
|---|
| 5 | 70 | 1.414 |
|---|
| 6 | 169 | 1.4142 |
|---|
| 7 | 408 | 1.4142 |
|---|
| 8 | 985 | 1.41421 |
|---|
| 9 | 2,378 | 1.41421 3 |
|---|
| 10 | 5,741 | 1.41421 3 |
|---|
| 11 | 13,860 | 1.41421 35 |
|---|
| 12 | 33,461 | 1.41421 356 |
|---|
| 13 | 80,782 | 1.41421 3562 |
|---|
| 14 | 195,025 | 1.41421 3562 |
|---|
| 15 | 470,832 | 1.41421 35623 |
|---|
| 16 | 1,136,689 | 1.41421 35623 7 |
|---|
| 17 | 2,744,210 | 1.41421 35623 73 |
|---|
| 18 | 6,625,109 | 1.41421 35623 730 |
|---|
| 19 | 15,994,428 | 1.41421 35623 7309 |
|---|
| 20 | 38,613,965 | 1.41421 35623 73095 |
|---|
| 21 | 93,222,358 | 1.41421 35623 73095 0 |
|---|
| 22 | 225,058,681 | 1.41421 35623 73095 04 |
|---|
| 23 | 543,339,720 | 1.41421 35623 73095 048 |
|---|
| 24 | 1,311,738,121 | 1.41421 35623 73095 0488 |
|---|
| 25 | 3,166,815,962 | 1.41421 35623 73095 0488 |
|---|
| 26 | 7,645,370,045 | 1.41421 35623 73095 04880 |
|---|
| 27 | 18,457,556,052 | 1.41421 35623 73095 04880 |
|---|
| 28 | 44,560,482,149 | 1.41421 35623 73095 04880 1 |
|---|
| 29 | 107,578,520,350 | 1.41421 35623 73095 04880 1
|
|---|
| 30 | 259,717,522,849 | 1.41421 35623 73095 04880 16 |
|---|
| 31 | 627,013,566,048 | 1.41421 35623 73095 04880 168 |
|---|
| 32 | 1,513,744,654,945 | 1.41421 35623 73095 04880 168 |
|---|
| 33 | 3,654,502,875,938 | 1.41421 35623 73095 04880 1688 |
|---|
| 34 | 8,822,750,406,821 | 1.41421 35623 73095 04880 16887 |
|---|
| 35 | 21,300,003,689,580 | 1.41421 35623 73095 04480 16887
2 |
|---|
| 36 | 51,422,757,785,981 | 1.41421 35623 73095 04480 16887 2 |
|---|
| 37 | 124,145,519,261,542 | 1.41421 35623 73095 04480 16887 24 |
|---|
| 38 | 299,713,796,309,065 | 1.41421 35623 73095 04880 16887 242 |
|---|
| 39 | 723,573,111,879,672 | 1.41421 35623 73095 04880 16887 2420 |
|---|
| 40 | 1,746,860,020,068,409 | 1.41421 35623 73095 04880 16887 24209 |
|---|
| Sequence | Generators |
|---|
| 41 | 4,217,293,152,016,490 |
|---|
| 42 | 10,181,446,324,101,389 |
|---|
| 43 | 24,580,185,800,219,268 |
|---|
| 44 | 59,341,817,924,539,925 |
|---|
| 45 | 143,263,821,649,299,118 |
|---|
| 46 | 345,869,461,223,138,161 |
|---|
| 47 | 835,002,744,095,575,440 |
|---|
| 48 | 2,015,874,949,414,289,041 |
|---|
| 49 | 4,866,752,642,924,153,522 |
|---|
| 50 | 11,749,380,235,262,596,085 |
|---|
| 51 | 28,365,513,113,449,345,692 |
|---|
| 52 | 68,480,406,462,161,287,469 |
|---|
| 53 | 165,326,326,037,771,920,630 |
|---|
| 54 | 399,133,058,537,705,128,729 |
|---|
| 55 | 963,592,443,113,182,178,088 |
|---|
| 56 | 2,326,317,944,764,069,484,905 |
|---|
| 57 | 5,616,228,332,641,321,147,898 |
|---|
| 58 | 13,558,774,610,046,711,780,701 |
|---|
| 59 | 32,733,777,552,734,744,709,300 |
|---|
| 60 | 79,026,329,715,516,201,199,301 |
|---|
| 61 | 190,786,436,983,767,147,107,902 |
|---|
| 62 | 460,599,203,683,050,495,415,105 |
|---|
| 63 | 1,111,984,844,349,868,137,938,112 |
|---|
| 64 | 2,684,568,892,382,786,771,291,329 |
|---|
| 65 | 6,481,122,629,115,441,680,520,770 |
|---|
| 66 | 15,646,814,150,613,670,132,332,869 |
|---|
| 67 | 37,774,750,930,342,782,945,186,508 |
|---|
| 68 | 91,196,316,011,299,234,022,705,885 |
|---|
| 69 | 220,167,382,952,941,249,990,598,278 |
|---|
| 70 | 531,531,081,917,181,734,003,902,441 |
|---|
| 71 | 1,283,229,546,787,304,717,998,403,160 |
|---|
| 72 | 3,097,990,175,491,791,170,000,708,761 |
|---|
| 73 | 7,479,209,897,770,887,057,999,820,682 |
|---|
| 74 | 18,056,409,971,033,565,286,000,350,125 |
|---|
| 75 | 43,592,029,839,838,017,630,000,520,932 |
|---|
| 76 | 105,240,469,650,709,600,546,001,391,989 |
|---|
| 77 | 254,072,969,141,257,218,722,009,304,910 |
|---|
| 78 | 613,386,407,933,224,037,990,008,001,809 |
|---|
| 79 | 1,480,845,785,007,705,294,702,019,308,528 |
|---|
| 80 | 3,575,077,977,948,634,627,394,046,618,865 |
|---|
| 81 | 8,631,001,740,904,974,549,490,112,546,258 |
|---|
| 82 | 20,837,081,459,758,583,726,374,271,711,381 |
|---|
| 83 | 50,305,164,660,422,142,002,238,655,969,020 |
|---|
| 84 | 121,447,410,780,602,867,730,851,583,649,421 |
|---|
| 85 | 293,199,986,221,627,877,463,941,823,267,862 |
|---|
| 86 | 707,847,383,223,858,622,658,735,230,185,145 |
|---|
| 87 | 1,708,894,752,669,345,122,781,412,283,638,152 |
|---|
| 88 | 4,125,636,888,562,548,868,221,559,797,461,449 |
|---|
| 89 | 9,960,168,529,794,442,859,224,531,878,561,050 |
|---|
| 90 | 24,045,973,948,151,434,586,670,623,554,583,549 |
|---|
| 91 | 58,052,116,426,097,312,032,565,778,987,728,148 |
|---|
| 92 | 140,150,206,800,346,058,651,802,181,530,039,845 |
|---|
| 93 | 338,352,530,026,789,429,336,170,142,047,807,838 |
|---|
| 94 | 816,855,266,853,924,917,324,142,465,625,655,521 |
|---|
| 95 | 1,972,063,063,734,639,263,984,455,073,299,118,880 |
|---|
| 96 | 4,760,981,394,323,203,445,293,052,612,223,893,281 |
|---|
| 97 | 11,494,025,852,381,046,154,570,560,297,746,905,442 |
|---|
| 98 | 27,749,033,099,085,295,754,434,173,207,717,704,165 |
|---|
| 99 | 66,992,092,050,551,637,663,438,906,713,182,313,772 |
|---|
| 100 | 161,733,217,200,188,571,081,311,986,634,082,331,709 |
|---|
| 101 | 390,458,526,450,928,779,826,062,879,981,346,977,190 |
|---|
| 102 | 942,650,270,102,046,130,733,437,746,596,776,286,089 |
|---|
| 103 | 2,275,759,066,655,021,041,292,938,373,174,899,549,368 |
|---|
| 104 | 5,494,168,403,412,088,213,319,314,492,946,575,384,825 |
|---|
| = | 1.41421 35623 73095 04880 16887 24209 69807
85696 71875 37694 80731 76679 73799 07324 78462 10703 |
|---|
Table 2. The First Ten generators of the sequence of Pythagorean Triples converging to a 30,60,90 right triangle
| Sequence | Generator | (3**(1/2))-2 | right triangle |
|---|
| n | k(n)=4k(n-1)-k(n) | (k(n)/k(n-1))-2 | side,side,diagonol |
|---|
| 0 | 1 |
|---|
| 1 | 4 | 2 | 15, 8, 17 |
|---|
| 2 | 15 | 1.75 | 209, 120, 241 |
|---|
| 3 | 56 | 1.73 | 2911, 1680, 3361 |
|---|
| 4 | 209 | 1.732 | 40545, 23408, 46817 |
|---|
| 5 | 780 | 1.73205 | 564719, 326040, 652081 |
|---|
| 6 | 2911** | 1.732051 | 786552, 4541160, 9082321 |
|---|
| 7 | 10864 | 1.7320508 | 109552575, 63250208, 126500417 |
|---|
| 8 | 40545 | 1.73205087 | 1525850529, 880961760, 1761923521 |
|---|
| 9 | 151316** | 1.732050875 | |
|---|
| 10 | 564719 | 1.7320508756 | |
|---|