1. ## A small problem about primes number

Conjecture.
Does there exist a real number C such that for all i in N:
tan(p_i)+cot(p_(i+1)) > 0 or tan(p_i)+cot(p_(i+1)) < C,
where C = -0.14627836...

If the conjecture was true, what is the exact value of C?

Best Regards,
Roberto Lepera

2. What are tg and ctg?

3. Originally Posted by Bruno J.
What are tg and ctg?
tg and ctg are tangent and cotangent functions

4. What makes you conjecture this?

I doubt anyone is going to be able to tell you whether that is true or not.

5. Originally Posted by rrronny
Conjecture.
Does there exist a real number C such that for all i in N:
tan(p_i)+cot(p_(i+1)) > 0 or tan(p_i)+cot(p_(i+1)) < C,
where C = -0.14627836...

If the conjecture were true, what is the exact value of C?

Best Regards,
Roberto Lepera
You've phrased it as a question, so I'm not sure what the conjecture is. Is this what you intend to state:
There EXISTS a real number C (with conditions)

It is not clear but for example:

$p_i$ is some prime such as 7
&
$p_{i+1}$ is the next prime or 11

Is that correct?

--
The $\tan$ & $\cot$ are reciprocals. When one of the two is greater than a unit the other will be a fraction. The sum will always exceed a unit.

When $p_i$ AND $p_{i+1}$ are reduced to 0 < p < $2\pi$
THEN

the sum will always be greater than 1.

the sum will always be less than -1.

the sum will always be less than -1.

the sum will always be less than -1.

When the tan--cotan pair occur in the 1st & 2nd quadrant or when they occur in the 4th and 1st quadrant will the sign of the two be different. But one will be exceedingly small and the other exceedingly large, since they are reciprocals.

What am I missing here?

From this quick look the value of 'c' is:

c < -1
or
+1 < c

Is there a C , such that -1 < c < +1
Don't think so.

Could you elaborate, and give a specific example so that your speculation can be fully understood?

6. The conjecture is true for first 200.000.000 prime numbers.
Code:
K = 10^8; c = -0.146278362977478; For[i=1, i≤K, i++,
{q = Tan[Prime[i]] + Cot[Prime[i + 1]]; If[q>c && q<0, Print[{i, N[q, 15]}]]}]

7. The conjecture is false for i=886792302.

8. Originally Posted by rrronny
Code:
K = 10^8; c = -0.146278362977478; For[i=1, i≤K, i++,
{q = Tan[Prime[i]] + Cot[Prime[i + 1]]; If[q>c && q<0, Print[{i, N[q, 15]}]]}]
After seeing the code, the meaning of the post became clear. A long while ago there was a problem about the gap between prime numbers. I have searched but have not been able to find the source of the information.

Tom Nicely has a fantastic web site about the gap between prime numbers.
I do not have it handy (you may use google to find it).

Back to the problem.
Roughly it was something like this:

A is some odd number (not required to be prime).
B is another odd number (or A + 2k, k being an natural number).

tan(A) = $\frac {sin A}{cos A}$
tan(A) + cotan(A + 2k) = c

that becomes:

$\frac { cos (2k) }{sin(A)cos(A)}$

A & 2k are integer values of radians. The sin/cos values will never be zero or one.

$0 < cos (2k) < 1$ & $0 < \sin(A) \cos(A) < 1$

The question was how close to zero (or one or negative one) does this get?
[Actually no integer values ever occur.]

Your question then becomes, does being a prime number of radians create a special characteristic of the limits?

This was answered in part by the indication that 2k (the gap between the prime numbers) needs to be
710
207986
208696
416682
625378
1667438
and so forth.
The gap between the prime numbers required increases somewhat rapidly.

See Tom Nicely's web page for the size of primes with those gaps.

The conjecture is false for i=886792302
I do not know the values of
$Prime_{886792302}$ & $Prime_{886792303}$ but the gap is the critical factor.

9. What I hardly undestand is the role of the constant c= - .146278362977478... Why this value and not any other value? ...

Kind regards

$\chi$ $\sigma$

10. The previous message has be sent two times... Very sorry!... I please a moderator to cancel this last post!... Thank you very much!...

Kind regards

$\chi$ $\sigma$

11. I know only that:
C=2*tan(x*), where x* is the first root negative of the equation tan(x)*tan(x+8)=1.

Best Regards,
R.L.

12. Originally Posted by rrronny
Since I’m Italian I suppose to be able to speech Italian!… Regard to that some time ago somebody in mathelpforum asked me about my avatar. Now I have a good opportunity to answer to You and Him in only one post…

The avatar is the reduction of this image...

This stone plate is set at the main entrance of 'Collegio Ghislieri' in Pavia [Italy] , the college in which I followed my university studies a lot of years ago. For information about the college see...

http://www.ghislieri.it/index.php

The sentence on the plate is taken from the 'Divina Commedia' of Dante Alighieri, the most famous italian poet, and in english can be translated as...

'... the more you know, the less you like wasting time...'

The avatar is a [negative] detail of this photo...

... that my wife did an year ago at the annual college party...

Regarding the ‘interesting’ problem You have proposed my opinion is that if in www.scienzematematiche.it there is a lot of people who love wasting time… the same is not here, so that please remember the sentence of Dante before propose to us an other problem…

Kind regards

$\chi$ $\sigma$