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of positive integers is called central if for every positive integer 
, the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.
+3 w
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
be positive real numbers such that : 
. Prove that : 
such that 
is a prime or a perfect square of an integer.
.
(prime).
But we know that 
. Contradiction.
.
(mod 5) . But 
(mod 5) 
Contradiction unless .
is the only solution.
. Viewing the equation modulo 
, we see that there are no possible solutions.
(
is a prime)
(for 
), we see that 
if an integer is always 
\for . works and thus it is the only possible solution.
, which clearly works. Then, we claim that no such 
exists.
(mod 4)
(mod 4)
to be a prime.
(mod 5), except .
only, which clearly works.
worked. Then, clearly we must have 
.
This is not a QR mod 
,
.
for some 
,
For 
and 
are both greater than 
,
,
which is the product of two integers greater than 
,
,
which is again the product of two integers greater than 
: No such n exists such that the ![\[ \]](http://latex.artofproblemsolving.com/6/1/1/611ad9d590175c01c2041fe506310cb1c96ef993.png)
Say there is a ![\[ 5^n - 1 \equiv 3p \pmod{4} \]](http://latex.artofproblemsolving.com/2/1/5/215f60a1a41c19a4ca7625ba67fc0a839e73dd9c.png)
![\[ p \equiv 0 \pmod{4} \]](http://latex.artofproblemsolving.com/4/5/1/451b9434003ee6028beb13d1ef03e6c3bf2c7812.png)
then![\[ 5^n-1 \equiv 3x^2 \pmod{5} \]](http://latex.artofproblemsolving.com/e/b/4/eb43529ee1dbf00dc59d011b3d3a6376eb700eb4.png)
![\[ -1 \equiv 3x^2 \pmod{5} \]](http://latex.artofproblemsolving.com/b/9/5/b950a028d7661edaf24eacb5a35aad5f8663afcb.png)
Thats leads us with no sol for 
.
follows.
and 
and hence it cannot be a prime or a perfect square.
is an integer; therefore, we need 
to either be of the form 
for a natural number 
or of the form 
for a prime number 
,
is a quadratic non-residue 
- contradiction!
,
,
No such