Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

James P. Jones and Péter Kiss: Properties of The Least Common Multiple Function

LEMMA

23. L{x) divides x\. PROOF. Induction on x, using Lemm 2.2. from L(x + \)\L(x)(x + \) and L(x)|x!, L(X + 1)|X!(X + 1). follows. DEFINITION 2.4. O(x) = L(x) LEMMA 2.5. Q(x) is an integer and Q(x) ] x

!. PFOOF. By Lemma 2.3. LEMMA 2.6. Q(x +1) = (ß(x) (x +1), xl). FROOF. From Lemma 2.1. using g(x +

1) = ö(x).(l(x),x + l). LEMMA 2.7. p is a prime if and only if L(p) = p L(p 1). LEMMA 2.8. p is a prime if and only if Q(p) = Q(p 1). DEFINITION 2.9. K(x) = . LEMMA 2.10. A'(x) is an integer, ^(x) = (L(x- l),x) and K(x)\x. FROOF. Use Lemma 2.6. LEMMA 2.11. p is prime iff K(p) = 1. LEMMA 1.12. p is composite iff i < K(p). 71