A TABLE OF PARTITIONS INTO TWO SQUARES WITH AN APPLICATION TO RATIONAL TRIANGLES.

Transcription

1 MATHEMATICS A TABLE OF PARTITIONS INTO TWO SQUARES WITH AN APPLICATION TO RATIONAL TRIANGLES. BY A. VAN WIJNGAARDEN (Report R 68 of the Computation Department of the Mathematical Centre at Amsterdam) (Communicated by Prof. D. VAN DANTZIG at the meeting of April 29, 1950) 1. Introduction. In order to solve a specific problem on rational triangles, mentioned in section 3, a table was needed of all partitions of a nu mb er n into two squares according to (1. 1) for n up to many thousands. C. E. BICKMORE and O. WESTERN 1) gave such a table but only for n up to 1000, whereas M. RIGNAUX 2) announced a manuscript table up to n = that is, however, not published to our knowiedge. Therefore, such a table was prepared that is reproduced here in full. 2. Preparation ot the table. It is well known, which numbers n can be written in the form (1. 1). In fact they are exactly the numbers that have no prime factors of the form 4k + 3 to an odd power. Also the number of partitions can be determined without difficulty for each n whose factors are known but the determination of the actual partitions, though straightforward, is rather tedious. In constructing a table of some size, one better starts, therefore, from the right hand side of (1. 1). On a National accounting-machine an auxiliary double-entry table of n was made by building up from a constant second difference for p2 + q2 ~ Twelve columns were produced simultaneously. From this auxiliary table the wanted one was constructed by rewriting the partitions p, q as function of n. This table was first checked against omissions by counting the number of partitions in both tables between round values of n, and next against reproduction errors by verifying that (1. 1) holds for all entries. The same checks were applied to the manuscript for the printer (that had to be arranged in another way) and to the proofsheets. The tab Ie is arranged in six lines of three columns each. The first column gives n, the second and third pand q. In case of double and multiple partitions the argument n is not repeated in order to facilitate their recognition. 1) C. E. BICKMORE and O. WESTERN, Messenger Math. 41, (1911). 2) M. RIGNAUX, L'intermédiaire des math., 25, 143 (1918), 26, (1919).

2 An application of the table in the field of rational triangle8. An application of this tabie, which was in fa ct the immediate cause for its construction, is the finding of the "smal1est" triangle (a, b, c) with integer sides a, band c and three rational medians. Here, by definition, a triangle (a, b, c) is smaller than another triangle (A, B,O), where a ~ b ~ c, and A<;B<;O, ifeitherc <Oorc=Oandb<Borc=O, b=banda<a. L. EULER 3 ) has given already some examples of these triangles, the smallest of which is (68, 85, 87). Several others have been found later on and by following the method of J. H. J. ALMERING 4 ) arbitrarily many can be constructed without difficulty. This method, however, does not permit to state what is the smallest solution, but the fact that nobody ever found a sma.]ler triangle than EULER'S one, mentioned above, induces the hypothesis, that it is indeed the smallest one. Of course, for the prove of this hypothesis, it has only to be verified th at lio smaller triangle than (68, 85, 87) has three rational medians, but the amount of smaller triangles is excessive. It shall be shown, however, that by the use of the table this verification is only a minor eomputing job. Be A, Band 0 the double medians on the sides a, band c respectively (so that c and 0 are diagonals of the parallelogram with sides a and b. etc.). They follow from (3. 1) (3. 2) (3. 3) 2a 2 + 2b 2 - c 2 = 0 2, 2a 2 - b 2 + 2c 2 = B2, -a 2 + 2b 2 + 2c 2 = A 2 If a, band c are all odd, or if a and b are even, and c is odd, then 0 2 = 3 (mod 4), so 0 is irrational. If a, band c are all even, af2, bf2, cf2 are integers and define a smaller triangle. Without loss of generality it ean, therefore, be assumed that a and bare odd and c is even. Then (3. 1) ean be written in the form (3.4) ( -a + b)2 (a + b)2 _ (~)2 (Q\2 = '2) n, wh ere all terms between brackets are integers. Moreover, the inequalities of the triangle require (- a + b)f2 < cf2 < (a + b)f2. Hence, n is a number permitting a double partition, and possible values of n are found directly by inspection of the table. As for the smallest triangle eertainly holds a ~ 85, b ~ 87, n = (a 2 + b 2 )f2 ~ If a number n to be investigated has the partitions n = p2 + q2 = = r , with p < q, r < 8 and p < r < q then is also p < 8 < q. Therefore, each double partition with r =1= 8 gives rise to two triangles, viz. (- p + q, p + q, 2 r) and (- p + q, p + q, 28). 3) L. EULER, Novi Cornm. Acad. Petrop., 18, 171 (1773). ') J. H. J. ALMERING, "Rationaliteitseigenschappen in de vlakke meetkunde", Thesis (1950), Amsterdam.

3 871 Not all double partitions come into consideration, however. As a and b are odd, n is also odd. In particular, the case r = 8 cannot occur. Also cases where p, q, r, 8 have a factor in common may be rejected. In this way only a rather limited amount of values of n need a closer investigation. For those n the corresponding a, band c were computed and inserted into (3. 2). In the few cases that B2 proved to be the square of an integer, they were also inserted into (8. 3), and in only one case A2 was the square of an integer, viz. in that of EULER'S triangle (68, 85, 87). This verification was checked by complete duplication. Hence, the smallest triangle with integer sides and three rational medians is (68, 85, 87). 4. Acknowledgement. I am indebted to Mr H. DUPARC for useful suggestions, to Miss G. BOTTERWEG, MissP. M. HEINSEN and Miss R. D. M. MULDER for performing the necessary calculations and to Miss C. LANGEREIS for the preparation of the manuscript.

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