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Soc. Transl. (2), 19 (1962), 299–321. [5] A. Schinzel, Ungelöste Probleme, Nr. 30. Elem. Math. 14 (1959), 60–61. [6] Y. Wang, On the representation of large integer as a sum of prime and an almost prime. Sci. Sinica 11 (1962), 1033–1054. Originally published in Colloquium Mathematicum LXVIII (1995), 55–58 Andrzej Schinzel Selecta On integers not of the form n − ϕ(n) with J. Browkin (Warszawa) W. Sierpi´nski asked in 1959 (see [4], pp. 200–201, cf. [2]) whether there exist infinitely many positive integers not of the form n − ϕ(n), where ϕ is the Euler function.
Let 1 2 (p − 1) = q1α1 q2α2 · · · qsαs , 892 G. Arithmetic functions where qi (1 s) are different primes and αi i s 1− i=1 1 qi > 1− 1 1. Clearly s < 20, and 20 p 1/20 >1− 20 > 1 − ε. p 1/20 On the other hand, s 1− i=1 −1 1 qi σ 1 2 (p − 1) 1 2 (p − 1) ϕ s 1 2 (p − 1) 1 2 (p − 1) 1− i=1 1 . qi It follows that (1 − ε)−1 > σ 1 2 (p − 1) 1 2 (p − 1) 1 2 (p − 1) 1 2 (p − 1) ϕ > 1 − ε. In view of (7), this completes the proof. Proof of the Theorem. We begin with formula (1). For any ε > 0 we take a prime r > 1 + ε −1 and put a = r in Lemma 1.
The numbers γi = qi (i = 1, 2, . . , s), as different rational primes and the number l satisfy the conditions of the Lemma. Then there exists an infinite number of prime ideals p of the field Γ (ζl ), the degree of which is 1 and for which we have qj qi qν −α = 1 (ν = i, j ), = ζl j , = ζlαi , c p p p c whence c a = 1, p b α β −β α = ζl i j i j = 1. p In both considered cases, therefore, there exists such a rational prime l that the field b = 1, but Γ (ζl ) contains infinitely many prime ideals p, of the degree 1, for which p x a a = 1, whence = 1, then the congruence a x ≡ b (mod p) is insoluble.