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**1 - 3**of**3**### Residue Classes Having Tardy Totients

, 2008

"... We show, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n = nk of the congruence φ(n) ≡ ak (mod mk) exists and satisfies log nk/log mk → ∞ as k → ∞. Here, φ(n) is the Euler function. This answers a question raised in [3]. We also s ..."

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We show, in an effective way, that there exists a sequence of congruence classes ak (mod mk) such that the minimal solution n = nk of the congruence φ(n) ≡ ak (mod mk) exists and satisfies log nk/log mk → ∞ as k → ∞. Here, φ(n) is the Euler function. This answers a question raised in [3]. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function λ(n) and the least such n satisfies n ≪ m 13. 1

### PRODUCTS IN RESIDUE CLASSES

, 708

"... Abstract. We consider a problem of P. Erdős, A. M. Odlyzko and A. Sárkőzy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some ..."

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Abstract. We consider a problem of P. Erdős, A. M. Odlyzko and A. Sárkőzy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results “on average ” over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results. 1.

### On multiplicative congruences

, 807

"... Let ε be a fixed positive quantity, m be a large integer, xj denote integer variables. We prove that for any positive integers N1,N2,N3 with N1N2N3> m 1+ε, the set {x1x2x3 (mod m) : xj ∈ [1,Nj]} contains almost all the residue classes modulo m (i.e., its cardinality is equal to m+o(m)). We further s ..."

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Let ε be a fixed positive quantity, m be a large integer, xj denote integer variables. We prove that for any positive integers N1,N2,N3 with N1N2N3> m 1+ε, the set {x1x2x3 (mod m) : xj ∈ [1,Nj]} contains almost all the residue classes modulo m (i.e., its cardinality is equal to m+o(m)). We further show that if m is cubefree, then for any positive integers N1,N2,N3,N4 with N1N2N3N4> m 1+ε, the set {x1x2x3x4 (mod m) : xj ∈ [1,Nj]} also contains almost all the residue classes modulo m. Let p be a large prime parameter and let p> N> p 63/76+ε. We prove that for any nonzero integer constant k and any integer λ ̸ ≡ 0 (mod p) the congruence p1p2(p3 + k) ≡ λ (mod p) admits (1 + o(1))π(N) 3 /p solutions in prime numbers p1,p2,p3 ≤ N. 2000 Mathematics Subject Classification: 11L40 1 1