Wikipedia:Reference desk/Archives/Mathematics/2011 December 25

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December 25

Even numbers among intervals of primes

i conjecture that all even numbers can be found among the intervals of primes. ie, there is no value for n such that p+2n is composite for all p. has this been proven. ther are three stronger possible conjectures

1)the difference between consecutive primes would give all even numbers

2) all even numbers can be found an infinite number of times among theintervals of the primes

3) the difference between consecutive primes would give each even number an infinite number of times

what work has beeen done on any of these. thanks — Preceding unsigned comment added by 86.174.173.187 (talk) 11:48, 25 December 2011 (UTC)

See Polignac's conjecture. Gandalf61 (talk) 13:27, 25 December 2011 (UTC)

The following is only about computational results which prove nothing about larger intervals. If we allow probable primes as gap ends for consecutive primes then 38888 is the smallest even interval with currently no known occurrence in the tables of first known occurrence prime gaps at http://www.trnicely.net/index.html#TPG. My program was first to find many thousands of the intervals and could probably easily fill the hole at 38888 and many later holes. The program found a gap of exactly 100000.[1] Many of my original gaps have since been replaced by gaps by Michiel Jansen and others.

If we demand proven primes then in 2007 I proved all gap ends of all intervals up to 20000 which were listed with probable primes at the time. This proved there is an occurrence of all gaps below 20000. Since 2007 many of the gaps with proven primes have been replaced in the tables by gaps between smaller probable primes. I haven't bothered to prove those.

For intervals between non-consecutive proven primes, in 2004 I found an occurrence for all even intervals up to 10¹².[2] The largest initial prime required for this was 3307 for an interval of 496562420542. PrimeHunter (talk) 15:23, 25 December 2011 (UTC)

As Gandalf61 rightly said: see Polignac's conjecture. — Fly by Night (talk) 00:38, 26 December 2011 (UTC)

it doesnt talk about the other conjectures, or really provide any decent evidence for it. in other words, its a stub.

Arc length in polar form

Why is the arc length formula for a function in (r(θ),θ) form what it is on your page arc length and not simply $\int _{\theta _{0}}^{\theta _{n}}r(\theta )d\theta$? Since the formula for the arc length of a circle through an angle Δθ is rΔθ so shouldn't the arc length of a polar function be as close as desired to r_iΔθ_i through a small enough choice of Δθ_i / fine enough partition of the theta interval? And by adding all these r_iΔθ_i don't we have a normal Riemannian integral of r wrt theta? THanks. 24.92.85.35 (talk) 18:28, 25 December 2011 (UTC)

Your method assumes that r is constant over the interval Δθ, whereas in general it will vary. This cannot be ignored. Your method is a bit like measuring the length of diagonal ramp by approximating it as a staircase and then summing the lengths of all the treads. No matter how tiny you make the steps, you will never get the right answer. 86.179.114.20 (talk) 18:46, 25 December 2011 (UTC)

If you go to the Arc length article and navigate to the Finding arc lengths by integrating section, then you'll find what you need to know. — Fly by Night (talk) 20:20, 25 December 2011 (UTC)

when can we say that we've found the root of an equation?

It's been some days that I've been having a discussion with a friend about a problem: assume we want to solve the equation f(x)=0(and we know what f(x) is). Now if we say x=f^-1(0) have we solved the equation? Well if not, what is its difference with the following solution for the equation below:

x²-2=0

x²=2

x=${\sqrt {2}}$, x=_${\sqrt {2}}$

because if one asks what is ${\sqrt {2}}$, we answer "it's the number whose square is 2", in other words, we really didn't solve the equation, we just chose a symbol for its answer!So this led me to the conclusion that we can only say that we solved an equation if we can present a way to approximate its answer with rational numbers, and that led me to the conclusion that if we can find a way to approximate the answer of an equation(newton's method and stuff like that), we can claim that we solved it (It's the case with ${\sqrt {2}}$ too, we can approximate ${\sqrt {2}}$ with rational numbers).Now since I'm not a mathematician, I know there's got to be something wrong with my conclusion, can anyone explain this subject for me?(and please don't ONLY link Wikipedia articles, I wanna understand it...)--Irrational number (talk) 19:14, 25 December 2011 (UTC)

In a sense it's true that "we just chose a symbol for the answer". There is no way to write sqrt(2) exactly using decimal notation (or any similar notation), so there isn't really any other choice. "solving" an equation normally means representing the answer using some fairly small set of standard operators and functions (which includes the square root function or, more generally, raising numbers to non-integral powers). The sorts of things that are allowed in a "solution" may vary depending on the context, and in more specialised areas new exotic functions may be defined as the root of some equation and then used in further results. Numerical approximations to an answer are fine for practical applications, but mathematically there's a world of difference between a numerical approximation and the mathematically exact answer. 86.179.114.20 (talk) 20:23, 25 December 2011 (UTC)

It's a very interesting idea. Think of a square with sides of length one. It's diagonal will have length √2. Just as there is a number whose square is 25 – namely five – there is a number whose square is two. Just because you can't write it down doesn't mean it doesn't exist. What about π? You can't write that number down, but it is a real number. Take a look at the article Completeness of the real numbers. — Fly by Night (talk) 20:46, 25 December 2011 (UTC)

Perhaps you've solved it when it's in its simplest possible form? so if you don't know fx then you're above solution would work, but if you do, you have to simplify it. — Preceding unsigned comment added by 86.174.173.187 (talk) 21:22, 25 December 2011 (UTC)

http://www.wolframalpha.com/input/?i=x^5%3Dx%2B1 says: x = root of x^5-x-1 near x = 1.1673. That's a nice way of expressing the useless exact solution together with the useful approximate solution. Bo Jacoby (talk) 07:04, 26 December 2011 (UTC).

well my problem is when we ask what is the square root of 25 they say five, but when we ask what is the square root of two, they say it's the square root of two!:D(irrational number)--81.31.188.252 (talk) 07:27, 26 December 2011 (UTC)

Yes, but √2 is just as real of a number as √25. The number √25 is the unique positive number that when multiplied by itself gives 25 as the answer, while √2 is the unique positive number that when multiplied by itself gives 2 as the answer. The only difference is that √25 has a simple form and can be written as 5, while √2 isn't so pretty. To save space, we write √2. Like I said: read Completeness of the real numbers. — Fly by Night (talk) 23:31, 26 December 2011 (UTC)

You have highlighted the difference between the algebraic and the analytic approaches to solving equations. The algebraic approach tells us that there are (at most) two solutions to x²= 2 and that they must sum to 0 (because the coefficient of x is 0). However, it does not tell us anything about where the solutions are in relation to other numbers because "nearness" or "distance" are not algebraic concepts. The analytic approach tells us that one solution can be found between 1 and 2 (because 1² - 2 = -1 and 2² - 2 = 2) and so the other solution must lie between -1 and -2. By convention we call the positive solution ${\sqrt {2}}$ and so the negative solutuon is $-{\sqrt {2}}$.Gandalf61 (talk) 12:05, 26 December 2011 (UTC)