Author:
Cerna Maguiña B. M.,Blas H.,López Solís V. H.
Abstract
Abstract
We consider a set of equations of the form Pj
(x, y) = (10x + mj
)(10y + nj
), x ≥ 0, y ≥ 0, j = 1,2,3, such that {m
1 = 7, n
1 = 3}, {m
2 = n
2 = 9} and {m
3 = n
3 = 1}, respectively. It is shown that if
(
a
(
p
j
)
,
b
(
p
j
)
)
∈
ℕ
×
ℕ
is a solution of the j’th equation one has the inequality
p
j
100
≤
A
(
p
j
)
B
(
p
j
)
≤
121
10
4
p
j
, where
A
(
p
j
)
≡
a
(
p
j
)
+
1
,
B
(
p
j
)
≡
b
(
p
j
)
+
1
and pj
is a natural number ending in 1, such that
{
A
(
p
1
)
≥
4
,
B
(
p
1
)
≥
8
}
,
{
A
(
p
2
)
≥
2
,
B
(
p
2
)
≥
2
}
, and
{
A
(
p
3
)
≥
10
,
B
(
p
3
)
≥
10
}
hold, respectively. Moreover, assuming the previous result we show that
1
≤
(
A
(
p
j
+
10
)
B
(
p
j
+
10
)
A
(
p
j
)
B
(
p
j
)
)
1
/
100
≤
e
0
,
000201
×
(
1
+
10
p
j
)
(
0
,
101
)
2
, with
{
A
(
p
1
)
≥
31
,
B
(
p
1
)
≥
71
}
,
{
A
(
p
2
)
≥
11
,
B
(
p
2
)
≥
11
}
, and
{
A
(
p
3
)
≥
91
,
B
(
p
3
)
≥
91
}
, respectively. Finally, we present upper and lower bounds for the relevant positive integer solution of the equation defined by pj = (10A + mj
)(10B + nj
), for each case j = 1, 2, 3, respectively.
Subject
General Physics and Astronomy
Reference8 articles.
1. Colloquium: Physics of the Riemann hypothesis;Schumayer;Reviews of Modern Physics,2011
2. The structure factor of primes;Zhang;J. Phys. A: Math. Theor.,2018
Cited by
1 articles.
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