Wikipedia:Reference desk/Archives/Mathematics/2009 October 21

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October 21

Extension of the Bateman-Horn conjecture to subsets

Is there an extension of the Bateman-Horn conjecture to give an expected measure of how many times k out of n polynomials that satisfy some kind of simple modified Bunyakovsky property jointly produce primes? I'm interested in getting a grip on the likely size of the first number x that satisfies the condition that seven of the first eight values in {x²ⁿ+xⁿ-1} are prime as well as the number less than a large value that are prime for seven out of the first nine, ten, eleven and twelve. User:PrimeHunter has provided various new data in search for best cases here, but right now x=10 stands by itself in having the formula produce a prime at n=1, 2, 3, 5, 6, 7 and 9. (A nine-digit example is the first giving primes for n=1 through 6, and it has the salutary additional property of giving a prime for n=10, with this latter being quite a nice little coincidence unlikely to be further repeated (for 'small' values). For x=10, n=13 and 26 give the next primes. This is an established sequence, as has been pointed out to me. We have 8 out of 15 to go along with 7 out of 10 for our second best cases (compared with 10 as best), so far. No search as yet has been endeavored to generate what should be easier to improve upon 10 for, meaning finding 9 out of 25 or better.Julzes (talk) 07:34, 21 October 2009 (UTC)

Ostrowski

Alexander Ostrowski appears to have been a reasonably productive mathematician through a long career that started in the 1920's. But he seems to have published a spurt of books in the 1980's, at a VERY advanced age (his late 80's or even older). Any idea how that came about? Were they works he had written earlier and finally published towards the end of his life? Or did he have a burst of mathematical activity as an octogenarian? I guess it's possible. 69.228.171.150 (talk) 10:52, 21 October 2009 (UTC)

I don't know what books he wrote, but mathematics textbooks are often based on the lecture notes the author has been using and improving for years. He might have decided when he retired from lecturing to publish his lecture notes (since he wasn't going to improve them any more). --Tango (talk) 11:25, 21 October 2009 (UTC)

I am really intersted in mathematitics adnd solving questions is one step ahead. However I've always got problems here and there as below

Points P[0,0], Q[1,7] and R[-7,-1] lie on a circle. Find (a)the equation of the circle (b)the center of the circle (c)the radius of the circle.

Posted by Duke Orenge Nyakundi, Kenya.email: email redacted 21/10/2009 16:31 —Preceding unsigned comment added by 41.204.168.3 (talk) 13:31, 21 October 2009 (UTC)

As our article on Circles mentions, the general equation for a circle is $\left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}$ where point (a,b) is the center of a circle and r is the radius. Since you know three points, you can substitute for x and y for each of the points and get a system of simultaneous equations. (e.g. for point Q, you would have $\left(1-a\right)^{2}+\left(7-b\right)^{2}=r^{2}$) Since you have three points, you have three equations and with three unknowns you can use a little algebra to solve they system for a, b and r. -- 128.104.112.179 (talk) 14:34, 21 October 2009 (UTC)

Another way with fewer equations to solve. If S is [-4,4], by symmetry the centre will lie on PS extended, at C [-a,a] say. CP^2 = CQ^2 will give you an equation in a, solve it to give the co-ords of the centre. I suggest that you then show what you can do to get the radius and equation of the circle, as we don't do homework without some evidence of understanding the problem. [I should have said that S is the midpoint of PQ, so its co-ords should be [-3,3] ]→86.132.161.236 (talk) 14:45, 21 October 2009 (UTC)

Wikipedia:Reference_desk/Archives/Mathematics/2009_October_13#Vectors_and_matrix_algebra_dot_products_et_al also discussed this problem. Rckrone (talk) 17:11, 21 October 2009 (UTC)

If you do not like solving systems of simultaneous equations, there is an alternative method of solving the problem. Here are the steps.

1. Pick a random point G with (random) coordinates (GX,GY)
2. Calculate all three distances between G and P,Q,R (called the distances GP,GQ,GR)
3. Select the longest and shortest distances (call them FAR and NEAR)
4. Calculate the difference between FAR and NEAR (call it DIFF = FAR - NEAR)
5. Move point G in the direction of the furthest point by a travel distance of DIFF / 2
6. Repeat step 2 to Step 5 until the numerical value of DIFF is so small that you do not give a damn any more
7. G is now located at the center of the (desired) circle
8. the radius is now the value of GP or GQ or GR (they should all be the same)

For extra brownie points, try and figure out why this algorithm works.....

PS: Someone told me my algorithm does not work. However there is a easy fix for it. I will leave it to the readers to figure out what the fix is. (Hint: the direction to move G) 203.41.81.1 (talk) 20:40, 21 October 2009 (UTC)

QUADRATIC EQUATION

HI, WHAT IS QUADRATIC EQUATION..?? —Preceding unsigned comment added by Rsrashmi31 (talk • contribs) 18:56, 21 October 2009 (UTC)

Quadratic equation. Staecker (talk) 18:58, 21 October 2009 (UTC)

... i.e. one where the highest power is two (or "squared"). Dbfirs 22:42, 21 October 2009 (UTC)

Please do not bold your questions (unless it was a mistake - for instance, if "Caps Lock" were on). It appears rude to some people, and does not emphasize the necessity for us to answer them (which, I presume, would be the main reason undermining why one would wish to bold his/her question). With regards to your question, a polynomial of degree 2 is often referred to as a "quadratic polynomial" (for instance, $x^{2}+2x+1$, $x^{2}+3x$ and $7x^{2}+3$ are quadratic polynomials whereas $x^{3}+2x^{2}+5x$ is not). A quadratic equation is of the form "$p(x)=0$" where p is a quadratic polynomial. Note that, contrary to popular belief, a "quadratic equation" is not a "mathematician's idol". --PST 02:48, 22 October 2009 (UTC)

Really? Popular belief has it the equation is a mathematicians' idol? I've never heard of that. I must be impopular. —130.237.45.207 (talk) 08:04, 23 October 2009 (UTC)

From a historical point of view, I think we can't deny that the solution of the equation of second degree is one of the big achievements of humanity --pma (talk) 08:56, 24 October 2009 (UTC)

Usually, the first thing that comes to a laymen's mind when "mathematics" is mentioned, is "quadratic equation" (or at least this is the case in my experience). Polynomial equations are certainly one of the most fascinating concepts; although seemingly basic, they are quite complex in nature (this is highlighted by Abel's impossibility theorem, for instance). Succintly, in my view, equations are only the beginning of vast amounts of theory (but nevertheless, I highly respect those mathematicians who did discover formulas for solving equations of particular degrees). --PST 13:40, 24 October 2009 (UTC)

If you have a quadratic equation you want to solve, the internal revenue service can help.[1] —Preceding unsigned comment added by 69.228.171.150 (talk) 05:53, 24 October 2009 (UTC)

History of Mathematical Thought on Base Ten

I'm wondering if anyone in mathematics or any notable person elsewhere has taken up the issue of coincidences limited to base ten. Writing elsewhere, I assumed I was the first to make such an issue (with an unfinished piece called "The Base-Ten Hypothesis" that was briefly in h2g2's Peer Review), but perhaps some comments have already seen the light of day before. The major coincidences I have noted before are 1) e=2.718281828..., 2) log₁₀2=0.30102999..., and 3) 1001=7*11*13 (yielding the nice well-known tests for divisibility by 7 and 13). I would also add the results on primality in the sequence 109, 10099, 1000999, ... (The 5th, 6th, 7th, 9th, 13th, 26th, ... terms are also prime, and right now 10 holds some records for simultaneity of primality in the sequence of sequences given by {x²ⁿ+xⁿ-1}, though Schinzel's hypothesis H leads us to expect that arbitrarily many of the first values of n may give a prime for some (infinitely many) numbers, though the search for them is laborious.) Thanks if anyone can point me to anything in the literature.Julzes (talk) 22:05, 21 October 2009 (UTC)

Have you read Mathematical coincidence? There are a few specifically "decimal" examples listed there. AndrewWTaylor (talk)

Yes, I'm supposedly (according to me) doing some editing of that. The first three I give here are simple, while the other is quite extreme. The article in question deals with my 2) above, but other than that and perhaps some dealing with measures the article gives examples that are either not as simple as the first three or not as extreme as the fourth. I have a complex gestalt of material on 365+1/4 also, but it seems unnecessary to use it at the outset in arguing the issue that I'm raising.

I know there has been awareness of the specifics given here, but I wonder whether anyone has actually focused any attention on them as perhaps being meaningful: There is a small number of small numbers, so natural selection can account for us coming to use a special base with coincidences, but the coincidences form a kind of limited evidence that it may not have. To get from this limited evidence to something more significant, one needs something like my gestalt on 365+1/4 and/or Bode's Law (if it doesn't have a naturalistic scientific explanation) for anything approaching scientific proof of the concept, but one can begin raising the issue with this base-ten stuff alone.Julzes (talk) 22:17, 22 October 2009 (UTC)