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
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
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:
$\displaystyle p_i $ is some prime such as 7
&
$\displaystyle p_{i+1} $ is the next prime or 11
Is that correct?
--
The $\displaystyle \tan $ & $\displaystyle \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 $\displaystyle p_i $ AND $\displaystyle p_{i+1}$ are reduced to 0 < p < $\displaystyle 2\pi$
THEN
1st quadrant:
the sum will always be greater than 1.
2nd quadrant:
the sum will always be less than -1.
3rd quadrant:
the sum will always be less than -1.
4th quadrant:
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?
The conjecture is true for first 200.000.000 prime numbers.
See also this code for Mathematica:
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) = $\displaystyle \frac {sin A}{cos A} $
tan(A) + cotan(A + 2k) = c
that becomes:
$\displaystyle \frac { cos (2k) }{sin(A)cos(A)} $
A & 2k are integer values of radians. The sin/cos values will never be zero or one.
$\displaystyle 0 < cos (2k) < 1 $ & $\displaystyle 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.
I do not know the values ofThe conjecture is false for i=886792302
$\displaystyle Prime_{886792302} $ & $\displaystyle Prime_{886792303} $ but the gap is the critical factor.
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.
P.S.: If you speak the italian language, see also here: http://www.scienzematematiche.it/for...st=0&sk=t&sd=a
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
$\displaystyle \chi$ $\displaystyle \sigma$