User manuel silva - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:35:09Z http://mathoverflow.net/feeds/user/1019 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59471/arithmetic-progressions-inside-polynomial-sets Arithmetic progressions inside polynomial sets Manuel Silva 2011-03-24T20:33:52Z 2012-07-13T12:38:50Z <p>There are at most 3 perfect squares in arithmetic progression (Fermat, Euler). It was shown in [1] that if $n>2$ there are no three term arithmetic progression consisting of nth powers.</p> <p>Take a <em>non-linear</em> polynomial $p(x)$ with integer coefficients and consider the polynomial set $A_p = p(\mathbb{N})$. Let $f(A)$ be the maximal length over all arithmetic progressions whose elements are in $A$. It is not hard to prove, reducing it to the perfect squares case, that if the $\deg(p)=2$ then $f(A_p)&lt;4$. </p> <p><strong>Question 1</strong>: Is it true that $f(A_p)$ is always finite if $\deg(p) >1$ ?</p> <p>It is clear that there are no infinite arithmetic progressions inside polynomial sets, but I do not see how to exclude the possibility of having arbitrarily large AP.</p> <p><strong>Question 2</strong>: Is there a uniform bound for $f(A_p)$ over all polynomials $p(x)$ with $\deg(p)=k$ ?</p> <p>For example, can we find $10$ terms in arithmetic progression whose elements belong to some polynomial set $A_p$ with $\deg(p)=3$?</p> <p>[1] H. Darmon and L. Merel, Winding quotients and some variants of Fermat’s Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.</p> http://mathoverflow.net/questions/10223/linear-equation-with-primes Linear equation with primes Manuel Silva 2009-12-31T03:33:26Z 2010-09-13T05:12:16Z <p>Is there an integer $n$ with an infinite number of representations of the form $n=2q-p$, where $p$ and $q$ are both primes?</p> <p>Given a positive integer $k>1$, I would like to know for which (if any) integers $n$ the linear equation $q-kp=n$ admits an infinite number of solutions, where $p$ and $q$ are primes. (I'm not including $k=1$ because it reduces to well know open problems, $k=1$ and $n=2$ would be the twin primes conjecture)</p> <p>The density of the prime numbers implies that at least there are integers $n$ with an arbitrarily large number of representations.</p> http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/27301#27301 Answer by Manuel Silva for Examples of common false beliefs in mathematics. Manuel Silva 2010-06-07T02:37:27Z 2010-06-07T02:37:27Z <p>Some people believe there is no "formula" for the nth prime number. Of course there are many such formulas, even though not very useful: <a href="http://mathworld.wolfram.com/PrimeFormulas.html" rel="nofollow">http://mathworld.wolfram.com/PrimeFormulas.html</a></p> <p>The reason given for disbelieving the existence of a "prime number formula" is also curious: "because the primes are unpredictable". This believe is in contradiction with the simple fact that anyone can come up with an easy algorihtm which gives the nth prime number. There is something mystical associate with this ill-defined term "formula".</p> http://mathoverflow.net/questions/6337/increasing-bijection increasing bijection Manuel Silva 2009-11-21T00:47:10Z 2009-11-21T21:07:03Z <p>Using the back-and-forth method we can construct an increasing bijection from the set of rational numbers to the set of of rational numbers except zero.</p> <p><a href="http://en.wikipedia.org/wiki/Back-and-forth_method" rel="nofollow">http://en.wikipedia.org/wiki/Back-and-forth_method</a></p> <p>I would like to have a "natural" bijection. The algorithm resulting from the back-and-forth method behaves rather chaotically. </p> <p>It would be nice for example to have an uniform bound on the number of steps needed to evaluate the image of any given rational number $a=\frac{p}{q}$. I'm note sure what should count as a "step" here, maybe adding or multiplying integers.</p> http://mathoverflow.net/questions/4582/upper-bound-on-the-area-of-a-midpoint-pentagon Upper bound on the area of a midpoint pentagon? Manuel Silva 2009-11-08T02:34:59Z 2009-11-19T21:25:13Z <p>Starting with a convex pentagon P, we define the "middle polygon" Q, whose vertices are the middle points of the sides of the initial pentagon. The ratio between the areas of this polygons seem to always satisfy: 1/2 &lt; area(Q)/area(P) &lt; 3/4</p> <p>The lower bound is easy to obtain. I don't see how to get the upper bound.</p> <p>This problem is equivalent to the following one. Just forget about the middle polygon for a moment. Start with a convex pentagon and consider also all his 5 diagonals. You will obtain a central pentagon. Prove that the area of the new central pentagon is less than the sum of the areas of the five small triangles which have a side adjacent to the sides of this central polygon.</p> http://mathoverflow.net/questions/59471/arithmetic-progressions-inside-polynomial-sets Comment by Manuel Silva Manuel Silva 2011-03-25T20:05:29Z 2011-03-25T20:05:29Z @Mark: The points do not have to be on a straight line. Take for example (1,1), (5,25) and (7, 49) which correspond to the 3-AP of squares 1, 25, 49. http://mathoverflow.net/questions/59471/arithmetic-progressions-inside-polynomial-sets/59504#59504 Comment by Manuel Silva Manuel Silva 2011-03-25T00:51:20Z 2011-03-25T00:51:20Z @Gerry: The polynomial degree is fixed. I have changed the original formulation to make it more clear. http://mathoverflow.net/questions/29926/3n-2m-pm-41-is-not-possible-how-to-prove-it Comment by Manuel Silva Manuel Silva 2010-06-30T02:09:56Z 2010-06-30T02:09:56Z What can be said about the integers with a representation of the form 3^n−2^m or 2^n−3^m? http://mathoverflow.net/questions/6511/examples-of-mathematics-motivated-by-technological-considerations/6540#6540 Comment by Manuel Silva Manuel Silva 2010-03-10T03:27:50Z 2010-03-10T03:27:50Z The first paragraph is rude. http://mathoverflow.net/questions/6337/increasing-bijection/6341#6341 Comment by Manuel Silva Manuel Silva 2009-11-21T16:26:28Z 2009-11-21T16:26:28Z I don't see even how to constuct your sequence $a_i$ meeting my criteria. Can we decide what is the nth decimal digit for $\sqrt 2$ in an uniformly bounded number of steps? http://mathoverflow.net/questions/5353/how-to-respond-to-i-was-never-much-good-at-maths-at-school/5361#5361 Comment by Manuel Silva Manuel Silva 2009-11-14T02:44:30Z 2009-11-14T02:44:30Z A possible variant would be: - What about reading? Did you felt the same way about it? Because both are extremely important to understand the world. http://mathoverflow.net/questions/4582/upper-bound-on-the-area-of-a-midpoint-pentagon Comment by Manuel Silva Manuel Silva 2009-11-08T18:32:26Z 2009-11-08T18:32:26Z Should this question be closed? As far as I understood Anton decided to close it without even understanding what was being asked. http://mathoverflow.net/questions/4582/upper-bound-on-the-area-of-a-midpoint-pentagon Comment by Manuel Silva Manuel Silva 2009-11-08T03:50:56Z 2009-11-08T03:50:56Z Your &quot;solution&quot; deals with the regular pentagon case. The question asked is for any convex pentagon.