General theory of Banach algebras by C. E. Rickart

By C. E. Rickart

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We turn t o hi m next . FIBONACCI ( ^ L E O N A R D O O F P I S A ) Fibonacci (1 1 80—1 240 ) i s shor t fo r Filio Bonacci whic h wa s ho w Leonardo of Pisa calle d himself , a member of the Bonacci Family. Hi s mos t famou s work s are 1. Libe r Abbac i (1 202 , revised 1 208 ) 2. Flo s (1 225 ) 3. Libe r Quadratoru m About 1 22 5 Frederic k I I hel d cour t a t Pisa , an d o n tha t occasio n Fibonacc i wa s presented t o him , an d i t appear s tha t Flos wa s presented t o th e Empero r then .

I t follow s a t onc e from th e abov e identit y tha t i f (p , q) i s a solutio n t o (P m ) an d (r , s) i s a solution t o (P n ), the n (pr ± Nqs,ps ± qr) i s a solutio n t o ( P m n ) . ps + qr) 20 V. S . VARADARAJA N We thin k o f thi s a s th e composition o f (p , q) an d (r , s). Thi s proces s wa s calle d the bhdvand b y Brahmagupta , th e wor d meanin g "production " i n Sanskrit . I t i s easy t o verif y tha t i f p, q, r, s ar e al l positiv e integers , the n (pr + Nqs,ps - h qr) i s different fro m th e previou s tw o (se e exercis e 2 below).

Suppos e first tha t bot h a and b are even. The n c 2 = a 2 + b 2 is even, whic h make s c even. S o 2 divide s al l thre e o f them , violatin g ou r assumptio n o f primitivity . O n th e othe r hand, suppos e tha t a and b are bot h odd . The n a 2 an d b 2 ar e bot h odd , s o that c 2 is even, showin g that c must b e even. Bu t a 2 an d b 2 must bot h leav e the remainde r 1 whe n divide d b y 4 s o tha t c 2 mus t leav e th e remainde r 2 whe n divide d b y 4 , a contradiction, becaus e a s c i s even , c 2 i s divisibl e b y 4 !

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