000RM.dvi
26.5 Apollonius circle: the circular hull of the excircles 717 Appendix: Three mutually orthogonal circles with given centers Gi ...
718 The excircles ...
Chapter 27 Figurate numbers 27.1 Triangular numbers .................. Thenth triangular number is Tn=1+2+3+···+n= 1 2 n(n+1). T ...
720 Figurate numbers Find the first few triangular numbers which are squares: k 12 3 4 5 6 7 nk 18 49 mk 16 35 Tn 1 36 1225 Pali ...
27.3 Pentagonal numbers 721 27.3 Pentagonal numbers .................. The pentagonal numbers are the sums of the arithmetic pro ...
722 Figurate numbers 27.4 The polygonal numbersPn,k............... More generally, for a fixedk, thek-gonal numbers are the sums ...
27.4 The polygonal numbersPn,k 723 27.4.1 Appendix: Solution of Pell’s equation (1) Letdbe a positive integer which is not a squ ...
724 Figurate numbers Exercise 1.Prove that every hexagonal number is a triangular number. 2.Find two triangular numbers (apart f ...
Chapter 28 Polygonal triples We consider polygonal numbers of a fixed shape. For a given positive integerk, the sequence ofk-gon ...
726 Polygonal triples 28.1 Double ruling ofS .................... The surfaceS, being the surface of revolution of a rectangular ...
28.2 Primitive Pythagorean triple associated with ak-gonal triple 727 Proposition 28.1.The two lines lying entirely on the hyper ...
728 Polygonal triples replace(u, v)by(u−r, v+q), in which both entries are odd). Clearly, the integerz 0 =puis such thatqz 0 − ...
28.4k-gonal triples determined by a Pythagorean triple 729 (p, q, r)corresponds toat most oneline onSassociated withk-gonal trip ...
730 Polygonal triples 28.5 Correspondence between(2h+1)-gonal and 4 h- gonal triples Letk 1 <k 2 be two positive integers≥ 5. ...
Chapter 29 Sums of consecutive squares 1 Sum of squares of natural numbers 2 Sums of consecutive squares: odd number case 3 Sums ...
802 Sums of consecutive squares 29.1 Sum of squares of natural numbers ........... Theorem 29.1. 12 +2^2 +3^2 +···+n^2 = 1 6 n(n ...
29.2 Sums of consecutive squares: odd number case 803 29.2 Sums of consecutive squares: odd number case .... Suppose the sum of ...
804 Sums of consecutive squares k =6, 8 , 10 , 13 , 17 , 18 , 20 , 21 , 22 , 26 , 27 , 30 , 32 , 34 , 35 , 37 , 40 , 41 , 42 , 4 ...
29.3 Sums of consecutive squares: even number case 805 29.3 Sums of consecutive squares: even number case .... Suppose the sum o ...
806 Sums of consecutive squares 7.The equation(E 37 ′ )has solutions (185, 2), (2257,261), and (2849, 330). From these we constr ...
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