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Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11 (2000), 33–35. AN INEQUALITY INVOLVING PRIME NUMBERS ¸ Laurentiu Panaitopol From Euclid’s proof of the existence inﬁnitely many prime numbers one can deduce the inequality p1 p2 · · · pn > pn+1 , where pk is the k-th prime number. Using elementary methods, Bonse proves in [1] that 2 p1 p2 · · · pn > pn+1 for n ≥ 4, and 3 p1 p2 · · · pn > pn+1 for n ≥ 5. ´ Stronger results of the same nature have been obtained by J. Sandor in [2]. For example 2 2 p1 p2 · · · pn > pn+5 + p [n/2] for n ≥ 24. Without the restrictions imposed by the use of elementary methods the pre- ´ cise determination of the margin from which the inequality holds, L. Posa [3] proves the following result: For all k > 1 there is an nk such that k p1 p2 · · · pn > pn+1 for all n ≥ nk . The aim of the present note is to improve this inequality. We recall two results due to Rosser and Schoenfeld [5]: 1 (1) pn ≤ n log n + log log n − for n ≥ 20 2 1991 Mathematics Subject Classiﬁcation: 11A41 33 34 ¸ Laurentiu Panaitopol and x x (2) π(x) > + for n ≥ 59, log x 2 log2 x where we denoted by π(x) the number of prime numbers not exceeding x. We will also use the following result due to G. Robin [4]: log log n − a (3) θ(pn ) > n log n + log log n − 1 + for n ≥ 3, log n where a = 2.1454 and θ(x) = log p, the sum being taken after primes p. p≤x All these results allow as to prove the following Theorem. For n ≥ 2 n−π(n) p1 p2 · · · pn > pn+1 . We begin by proving the following Lemma. For n ≥ 59 we have log log n − 0.4 log pn+1 < log n + log log n + . log n Proof. It is well known that log x ≤ x − 1 for x > 0, from which we get for x = 1 + 1/n that 1 log (n + 1) < log n + n and then 1 1 1 log log (n + 1) < log log n + = log log n + log 1+ < log log n + . n n log n n log n We apply the inequality (1) and for n ≥ 19 we get 1 log pn+1 < log (n + 1) + log log (n + 1) + log log (n + 1) − 2 1 1 + log log n 1 1 < log n + + log log n + log 1 + + 2 − n log n n log n 2 log n 1 log log n 1 1 1 < log n + + log log n + + + − . n log n n log n n log2 n 2 log n It remains to show that log n + 1 1 1 + log log n + − < log log n − 0.4, n n log n 2 that is log n + 1 1 + < 0.1, n n log n which holds for n ≥ 59. An inequality involving prime numbers 35 Proof of the theorem. For n ≥ 59 we use (2) and the Lemma. We have 1 1 log log n − 0.4 n−π(n) log pn+1 < n 1 − − log n + log log n + . log n 2 log2 n log n In order to prove the theorem it is enough to show, using (3), that 1 1 log y − 0.4 log y − a 1− − 2 y + log y < y + log y − 1 + , y 2y y y where y = log n > log 59. This last inequality is equivalent to 1 log y − 0.4 a − 0.9 < 1+ log y + , 2y y which is true, since a − 0.9 < 1.3 and log y > log log 59 > 1.4. The theorem is thus proved for n ≥ 59. It may be checked directly that the assertion in the statement also holds for 2 ≤ n ≤ 58. From the above result we immediately obtain an improvement of L. Posa’s ´ inequality. Corollary. For any integer k, k ≥ 1, and n ≥ 2k the following inequality holds: k p1 p2 · · · pn > pn+1 . Proof. The function f : N∗ → N, f (n) = n − π(n) in increasing. For n ≥ 2k, f (n) ≥ f (2k) = 2k − π(2k) ≥ k, since π(2k) ≤ k for k ∈ N∗ . REFERENCES 1. H. Rademacher, O. Toeplitz: The enjoyment of mathematics. Princeton Univ. Press, 1957. ´ ¨ 2. J. Sandor: Uber die Folge der Primzahlen. Mathematica (Cluj) 30 (53) (1988), 67–74. ´ ¨ 3. L. Posa: Uber eine Eigenschaft der Primzahlen (Hungarian). Mat. Lapok 11 (1960), 124–129. e 4. G. Robin: Estimation de la fonction de Tschebyshev θ sur le k-i`me nombre premier et grandes valeurs de la fonction ω(n), nombre des diviseurs premier de n. Acta. Arith. 43 (1983), 367–389. 5. J. B. Rosser, L. Schoenfeld: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–89. a Facultatea de Mathematic˘, (Received August 3, 1998) s Universitatea Bucure¸ti, Str. Academiei nr. 14, RO–70109 Bucharest 1, Romania