AN INEQUALITY INVOLVING PRIME NUMBERS by ddh19362

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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

								
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