Download AN INEQUALITY INVOLVING PRIME NUMBERS

Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. 11 (2000), 33–35.
AN INEQUALITY
INVOLVING PRIME NUMBERS
Laurenţiu 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
for n ≥ 5.
p1 p2 · · · pn > pn+1
Stronger results of the same nature have been obtained by J. Sandór 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 precise determination of the margin from which the inequality holds, L. Pósa [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]:
¶
µ
1
for n ≥ 20
(1)
pn ≤ n log n + log log n −
2
1991 Mathematics Subject Classification: 11A41
33
34
Laurenţiu Panaitopol
and
(2)
x
x
+
for n ≥ 59,
log x 2 log2 x
π(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]:
µ
¶
log log n − a
(3)
θ(pn ) > n log n + log log n − 1 +
for n ≥ 3,
log n
P
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 pn+1 < log n + log log n +
log log n − 0.4
.
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
µ
¶
³
´
¡
¢
1
1
1
log log (n + 1) < log log n +
n
= log log n + log
1+
n log n
< log log n +
n log n
We apply the inequality (1) and for n ≥ 19 we get
µ
¶
1
log pn+1 < log (n + 1) + log log (n + 1) + log log (n + 1) −
2
µ
¶
1
1
1 + log log n
1
< log n + + log log n + log 1 +
+
−
n
log n
n log2 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
which holds for n ≥ 59.
1
log n + 1
+
< 0.1,
n
n log n
.
An inequality involving prime numbers
35
Proof of the theorem. For n ≥ 59 we use (2) and the Lemma. We have
µ
¶µ
¶
¡
¢
1
log log n − 0.4
1
−
log
n
+
log
log
n
+
.
n−π(n) log pn+1 < n 1 −
log n 2 log2 n
log n
In order to prove the theorem it is enough to show, using (3), that
µ
¶µ
¶
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
µ
¶µ
¶
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. Pósa’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∗ 7→ 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. Sandór: Über die Folge der Primzahlen. Mathematica (Cluj) 30 (53) (1988), 67–74.
3. L. Pósa: Über eine Eigenschaft der Primzahlen (Hungarian). Mat. Lapok 11 (1960),
124–129.
4. G. Robin: Estimation de la fonction de Tschebyshev θ sur le k-ième 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.
Facultatea de Mathematică,
Universitatea Bucureşti,
Str. Academiei nr. 14,
RO–70109 Bucharest 1, Romania
(Received August 3, 1998)