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Czech-Polish-Slovak Match
1996 Czech and Slovak Match
1
1
Part of
1996 Czech and Slovak Match
Problems
(1)
p prime iff 1 of a+b-6ab+(p-1)/6 , a+b+6ab+(p-1)/6 \in Z
Source: Czech and Slovak Match 1996 P1
10/1/2017
Show that an integer
p
>
3
p > 3
p
>
3
is a prime if and only if for every two nonzero integers
a
,
b
a,b
a
,
b
exactly one of the numbers
N
1
=
a
+
b
−
6
a
b
+
p
−
1
6
N_1 = a+b-6ab+\frac{p-1}{6}
N
1
=
a
+
b
−
6
ab
+
6
p
−
1
,
N
2
=
a
+
b
+
6
a
b
+
p
−
1
6
N_2 = a+b+6ab+\frac{p-1}{6}
N
2
=
a
+
b
+
6
ab
+
6
p
−
1
is a nonzero integer.
prime
Integers
number theory