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1. What is Riemann zeta function.

Thanks very much for the explanation. But there are some doubts which persist for me . You say that I should use the definition of Riemann function on the positive real axis , and extentd the definition for complex numbers . But the thing is that I was only able to find the definition for...
2. What is Riemann zeta function.

Well , I think I have understood something wrong , because I can think of many examples of functions that are completely differential in a domain , but there exist more than 1 continuation of that function over a larger domain, which is still differentiable. eg. f (x) = 1/ x^3 and f(x) =...
3. Did Paul Cohen settle the Continuum Hypothesis?

I would like to ask from where did the axioms of ZFC come from in the first place ? Why is it that you consider the axioms of ZFC relevant , but adding a few more axioms to ZFC , just an exercise for set theorists ? I am not a mathematician , and have very limited knowledge of mathematics...
4. What is Riemann zeta function.

By the way can we not represent the Riemann zeta function over the larger domain with the help of some power series ?
5. What is Riemann zeta function.

Well , after looking at a couple of places , I came to know that the analytic continuation is a function that has the same value as the given function within the given functions domain , but is defined at points in a larger superset of the original domain too . Also that the analytic function is...
6. What is Riemann zeta function.

Could anyone tell me what is the Riemann zeta function. On Wikipedia , the definition has been given for values with real part > 1 , as : Sum ( 1 / ( n^-s) ) as n varies from 1 to infinity. but what is the definition for other values of s ? It is mentioned that the zeta function is the...
7. Euclid's Proof

In the proof , it is not that p1 , p2 , p3 have been replaced by powers of 2 . All that it is saying is that : p1.p2.p3.p4.....pn < 2 . (2^2) . (2^4) .... ( 2 ^ (2^n - 1 ) ). This is because it is assuming the theorem to be true for p1 , p2 .. upto pn .
8. F(x)≡0 (mod 15) Find all roots mod 15

why at all should mn divide f(x) ????? consider f(x) = 18 , thus possible values for m and n are 6 and 9 , but clearly 6 x 9 = 54 does not divide 18 , but had m and n been coprime then we would have a completely different answer.
9. Why doesnt Bertrand's postulate imply Legendre's conjecture?

It looks to me like your proof that legendre implies andrica , is implicitly assuming that there are 2 primes between n^2 and (n+1)^2 . But Legendre guarantees only the existence of a minimum of one. If there is only 1 prime bw n^2 and (n+1)^2 , then I am unable to follow how you conclude that...
10. A seemingly simple combinatorial problem

I don't get it , why should the answer be r+p-1Cp-1 , if the coefficients are different . Consider : a + b = 5 There are 5 nonnegative integer solutions Now consider : a + 2b = 5 the only solutions that a can take are 1,3,5 - only 3 solutions .
11. A couple of Infinite Series questions

Ok .. I got how to find the sum : \sum_{r= 1}^\infty \frac{1}{2r(2r+1)} \sum_{r= 1}^\infty \frac{1}{2r} - \sum_{r=1}^\infty \frac{1}{(2r+1)} Now consider the series expansion of log(1 + x ) and in this expansion...
12. A couple of Infinite Series questions

Thanks again .. what a fool I was .. Rather than considering the area under the curve from i to i+1 , I was considering the area under the curve from i-1/2 to i + 1/2 , thats why I had to take the difference and check for monotonicity and all ... it was pretty obvious . By the way Ubibic ...
13. A couple of Infinite Series questions

Thanks Robert for the explanation . Yes I was able to show that \sum_{i=1}^r 1/i > \int_1^r\frac{1}{r} =ln(r) But what bothers me is that I had formed some sort of argument in my mind to convince myself that if the integral is bounded for some function f(x) then the sum shall also...
14. A couple of Infinite Series questions

Well I just came to know that \sum_{r=1}^\infty 1/r is itself divergent , so I am unsure of your question , it looks like some - \infty + \infty problem . I am at loss to see whats goin on ... does it mean the sum is undefined (because it doesnt seem to be right to say that the sum is...
15. A couple of Infinite Series questions

If the sum is for values of r from 0 to infinity , then the 1st term in your second question is -infinity. So how is the sum supposed to converge ??
16. Number of Solutions to d(p)

Considering d(p,N) : take remainder of N divided by p . Let it be r. Now if r is a perfect square , then d(p.N) will have infinite solutions of the form r + kp else there is no solution
17. Number of Solutions to d(p)

I think you mean that N is fixed , but If the question is : Find all N < p such that n^2 = N ( mod p) for some n ( By some n , I mean that corresponding to each N there will be one n) , then : d(p) = greatest integer less than square root of p
18. Number of Solutions to d(p)

I dont think this is the right solution . Consider p = 3 and N = 2. We know that no square number is congruent to 2 modulo 3. So in this case d(3) = 0
19. Fundamental Theorem of Arithmetic Problem

I think you should pay attention to the fact t1 , t2, etc . are greater than OR EQUAL TO zero. This should solve your purpose.
20. Why doesnt Bertrand's postulate imply Legendre's conjecture?

I was able to see that Andrica's Conjecture does indeed lead to Legendre's Conjecture . Regarding the converse - i.e. given Legendre's Conjecture , then Andrica's Conjecture also holds . -- I can see that if Legendre,s Conjecture is stated for real values of n > 1 , and not just integer...
21. Why doesnt Bertrand's postulate imply Legendre's conjecture?

Ok ..... I was able to show that eq 2 ..i.e : log ( [x] ! ) = \Psi(x) + \Psi(x/2) + \Psi(x/3) + .... is indeed valid. But still this was not at all obvious to me , and only after reading the lemma 2 part in the wikipedia proof of Bertrands postulate (...
22. Why doesnt Bertrand's postulate imply Legendre's conjecture?

Regarding Ramanujan's proof of the Bertrand postulate , I am unable to understand the 2nd equation that he uses ...... He starts the proof like this , let v(x) be the sum of logarithms of all primes less than or equal to x, Now consider : \Psi(x) = v(x) + v (x^[1/2]) + v(x^[1/3]) + ...
23. Perfect square in 2 bases

What I am looking for is basically an algorithm ... Suppose I find one such base by exhaustive testing , now is it possible to find another base , or do I have to do exhaustive testing to find the second base too . Or is there some way to rule out a certain sequence of digits from being a...
24. Fermat primes

Well I really donno if ur heading in the right way .. but very nice thinking .... If I can be of any help , You can express N(i+1) in terms of N(i) .. ( that is if my calculation is correct ) as follows : N(i+k) = { (6^(3k)) * ( [ { N(i) + (2/3) } ^ 2 ] ^ {2k} ) } -...
25. How to prove this theorem

Well .. I think u got it absolutely right here ..... I was saying the same thing basically .. that x < \Pi\pi(n) +1 < x! .... but nice to see your proof of it ..... by the way thanx for ur example (for instance 2*3*5*7*11*13+1=30031= 59*509) .... coz i was searching for such an example but...
26. How to prove this theorem

I didnt understand the proof completely , but this is what I think u meant : you have proved that ∏p¡(n) + 1 + 1 > n and < n!, and then concluded that ∏p¡(n) + 1 is a prime number due to some theorem by euclid. Well if thats the case , then what I can say is ∏p¡(n) + 1 , need not be a prime...
27. How to prove this theorem

OwlHoot, A few questions here , how is it that if an integer == 1 mod each prime < n then that integer == 1 mod (n-1)! ?? eg. consider n = 6 , then the required integer = 2*3*5 + 1 ( becoz it gives rem 1 with all primes < 6) but 2*3*5 + 1 mod (6-1)! , ie mod 120 is certainly not == 1 . it...
28. Perfect square in 2 bases

Does anyone know if its possible to check whether a number is a perfect square in 2 different bases ? ( I dont mean the representation of a number in 2 different bases - coz that would be a ridiculous question , what I mean is consider a no say xyz - is it possible to prove / disprove that it is...