# Search results

1. ### Order of 3 modulo a Mersenne prime

Hi, I have the following (new, I think) conjecture about the Mersenne prime numbers, where: M_q = 2^q - 1 with q prime. I've checked it up to q = 110503 (M29). Conjecture (Reix): \large \ order(3,M_q) = \frac {M_q - 1}{3^O} where: \ \large O = 0,1,2 . With I = greatest i such that...
2. ### Three conjectures looking for a proof ! 100Euro reward !

Hi, I've put on the http://mersenneforum.org/showthread.php?t=10670" the description of 3 conjectures that are waiting for a proof. I've already done half the proof for one of them (the easy part...). I've provided PARI/gp code that exercises the 3 conjectures. I'll give 100Euro for the...
3. ### LLT numbers

Let's say: L(x)=x^2-2 , L^1 = L, L^m = L \circ L^{m-1} = L \circ L \circ L \ldots \circ L. Where L(x) is the polynomial used in the Lucas-Lehmer Test (LLT) : S_0=4 \ , \ S_{i+1}=S_i^2-2=L(S_i) \ ; \ M_q \text{ is prime } \Longleftrightarrow \ S_{q-2} \equiv 0 modulo M_q . We have...
4. ### Another (candidate) test of primality of Mersenne number

Hi, You probably already know the Lucas-Lehmer-Test (LLT) used for proving that a Mersenne number is prime or composite. (See: http://mathworld.wolfram.com/Lucas-LehmerTest.html" [Broken]). The LLT is based on the properties of the Tree built by x^2-2 modulo a Mersenne number. Now, here is a...
5. ### M43: GIMPS project has found a new Mersenne prime

Hi, The GIMPS project has found a new Mersenne prime: M43 ! \large 2^{30,402,457} - 1 It is the 43th known Mersenne prime and it is the new largest known prime. It has 9,152,052 digits ! So the BIG ONE (more than 10 millions digits) is still to be found ! (and its discoverer will...

Hi, I'd like to know which property proves the following simple result. Let p be a prime greatest than 3. r is a quadratic residue of p if there exists a such that: a^2 \equiv r \pmod{p}. Since p is prime, there are \frac{p-1}{2} different residues (not counting 0). Now, if you sum them...
7. ### A VERY interesting Fermat-like sequence: A_n=4^3^n+2^3^n+1

Hi, In 1995, Yannick Saouter produced the study of a family of numbers close to the Fermat numbers: A_n=4^{3^n}+2^{3^n}+1 . (See: http://www.inria.fr/rrrt/rr-2728.html) Saouter proved that this A_n serie shares many properties with the Fermat numbers: 3.4 A_n numbers are pairwise...
8. ### 2^n+1 = m^2

Hi, I'm looking to solutions of: 2^n+Q=m^2 , where Q=1 . Obviously, n must be odd. I already know the trivial solution: 2^3+1=3^2 and I've started using a naive PARI/gp program for finding (n,m) up to n=59 . No success yet. Do you know about other solutions or about some theory ? This is...
9. ### Help about properties of Pell numbers

Hi, I need some help about properties of Pell numbers: U_n = 2 U_{n-1} + U_{n-2} , \text{ with: } U_0=0 \text{ and: } U_1=1 V_n = 2 V_{n-1} + V_{n-2} , \text{ with: } V_0=2 \text{ and: } V_1=2 I have a proof for: \frac{V_{\displaystyle 2^{\scriptstyle n}}}{2} \ = \ 1 + 4...
10. ### Generalized Pell Numbers

Hi, In the following document: Generalized Pell Numbers, I've defined what I call "Generalized Pell Numbers". They provide a way for computing: 1+\sqrt[m]{m}. I'd like to know if these numbers are already known or not, and if someone knows about other properties they have or if someone is...
11. ### The GIMPS project has found a new Mersenne prime number: M42.

The GIMPS project is aimed to search (and find !) new Mersenne prime numbers. It has just discovered a new huge prime number, named M42, which has been verified with a different software on a different computer architecture (a third verification is on the way). It is the 8th Mersenne prime...
12. ### A primality test for Fermat numbers faster than Pépin's test ?

Hi, I've published on my site the following paper: "A primality test for Fermat numbers faster than Pépin's test ? Conjecture and bits of history" It is a kind of investigation about the history of Mathematics. http://tony.reix.free.fr/Mersenne/P...rmatNumbers.pdf [Broken] It is...
13. ### I can't add an avatar !

Hi, The Edit Avatar interface does not work (tried with IE 6.0 and Mozilla 1.6 and 1.7) for me. When I click the "No avatar" and the "Save Changes" buttons there is no additional menu that would enable me to provide my avatar. I already asked to the "Contact Us" email address, but got no...
14. ### I need a proof for this binomial property.

Hi, I've spent dozen of hours searching by my-self and dozen of hours searching on the Web. Now I need help. Who could provide a proof for this binomial property ? I need it for another proof. Thanks Tony Let: F_n=2^{2^n}+1 , n \geq 2 . Prove: F_n \text{ prime } \Longrightarrow F_n...
15. ### A property of Chebyshev polynomials

Hi, I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials: (T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [ Example : T_2(x) = 2x^2-1 T_3(x) = 4x^3-3x T_3(T_2(x)) = T_2(T_3(x)) =...