By W. Sierpinski

Because the booklet of the 1st version of this paintings, substantial growth has been made in lots of of the questions tested. This version has been up-to-date and enlarged, and the bibliography has been revised. the diversity of subject matters coated the following comprises divisibility, diophantine equations, leading numbers (especially Mersenne and Fermat primes), the fundamental mathematics services, congruences, the quadratic reciprocity legislation, enlargement of actual numbers into decimal fractions, decomposition of integers into sums of powers, another difficulties of the additive concept of numbers and the speculation of Gaussian integers.

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Hence, in virtue of (20) and the identity ab, -bal = da, b l -db l a l = 0 we obtain equality (19). Thus, In order that integers x and y constitute a solution of equation (19) it is necessary and sufficient that for some natural t formulae (23) hold. It follows that for t = 0, ± 1, ± 2, ... formulae (23) give all the integral solutions of equation (19). Since at least one of the numbers aI' b l is different from zero, if equation (19) has at least one integral solution, then it has infinitely many of them.

Taking into account non-primitive triangles with hypotenuses s; 37 we obtain other 8 triangles (6,8, 10), (9, 12,15), (12,16,20), (15,20,25), (10,24,26), (18, 24, 30),(30,16,34), (21,28, 35) of area 24, 54,96,150,120,216,240,294, respectively. Thus we see that there is no pair of triangles among the Pythagorean triangles with hypotenuses s; 37 such that both triangles ofthe pair have the same area, except the pair (21, 20, 29), (35, 12, 37). We note that two Pythagorean triangles of the same area and the equal hypotenuses are congruent.

The number YI - Yz can be a negative integer, but certainly it is different from zero, so Y = IYI - Y21 is a natural number. We see that X = Xl - Xz ~ q~ j;;, Y ~ q ~ j;; and that for the appropriate sign - the number a (Xl - X2) - (Yl - Yz) = ax ± Y is-divisible by m, and this is what the Thue theorem states. 0 ~ Xl + or ~ q~ By a slight modification of the proof given above we could have the following generalization of the theorem Scholz and Schoenberg proved ([1], p. 44): If m, e and f are natural numbers such that e ~ m, f ~ m < ef, then for each integer a with (a, m) = 1 there exist integers X and Y such that for the appropriate sign + or - we have m I ax ± y and 0 ~ X ~ 1, 0 ~ y ~ e.