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By Wieslaw Tadeusz Zelazko

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1 ) )= O and q ( n ! 1 ) g n ! 1 Next. 44 we have ( R ( n ! + l a)=O , and & DO +- whence q ( n ! + 1 ) is a prime number greater than n and not greater than n! + l . > 1 p ( x ) > m} -i x > O n>l)-+a>n ( R ( n ! +l))=O & q ( n ! , 1 + pl(m) 2 1 that is m>i +q ( m ) = i - ( q ( n r ) ~ l ) . TVriting ~ ( u L )= q ( n i )-I- l. so tliat m -. i then (for n 10). p(n) i s fhe nth odd prime number. -+ : y ( m ) . "{x>@(n) & p(x)=O) >p(o) & p(~p(m) 1) -- n i ] that is »? m) :) . Iii this and the foilowing section it is ciesirable to rc-iiitroducc t,he coiiveiitioiial sigil by whicli to espress the procluct of a.

B), ~ ( ab)l, by ( a , b) theii f ((1. O ) = O. j ( O . S b ) = 0 : and from q ( a , b ) = p ( S a . S b ) , y ( n . b ) = y ( S a . S b ) follows ! ( u . b ) = f ( S n . S b ) ; mhence f ( t i , b ) = O and so g(cc, b ) = yi(ci. b ) . -4s instaiices of this last sclieina \ve meiitioii \Ve observe first t,hat fmrii f ( O . O ) = O. f (O. ))= 0 follo~vsf ( U . h ) = O. The irnplicntion hypotliesis stniitls for the c:iiintioii Non- { l f (O, b - 1 ) ) f ( O . - 1); ! ( S u . b) = O . (u: 1, b 1): !

Is an illustration of the use of Sb, we derive F(n. b)= F ( A , B ) from the pair of equations a =A , b = B. First ure derive F(a, b)= F(n, B ) from b = E by Sb,, and similarly F(a, B ) = F ( A : B ) from a= A ; hence by using K and T we derive F ( a , b) = F ( A , B ) from a = A , b = B . Two further schemata of importante are To prove E,, we define Hl(x, t), C(t) explicitly by the asioms H,(x, t )=t C(t)= F ( 0 ) , wheilce we readily derive C(0)= F(O), C ( S x )= H,(x, C ( x ) ) ,F ( S x )= H,(z, F ( x ) ) which, by El,yields F(x)=C(x), and from this we reach F ( x )= F ( 0 ) by Sb,.

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