Goldbach–Euler theorem

Goldbach–Euler Theorem

Goldbach–Euler theorem

to get instant updates about 'Goldbach–Euler Theorem' on your MyPage. Meet other similar minded people. Its Free!

X 

All Updates


Description:
In mathematics, the Goldbach–Euler theorem (also known as Goldbach's theorem), states that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 and omitting repetitions, converges to 1:

<math>sum_frac= + cdots = 1.</math>


This result was first published in Euler's 1737 paper "Variae observationes circa series infinitas". Euler attributed the result to a letter (now lost) from Goldbach.

Proof

Goldbach's original proof to Euler involved assigning a constant to the harmonic series:<math> textstyle x = sum_^infty frac </math>, which is divergent. Such a proof is not considered rigorous by modern standards. It is also interesting to note that there is a strong resemblance between the method of sieving out powers employed in his proof and the method of factorization used to derive Euler's product formula for the Riemann zeta function.

Let x be given by

<math>x = 1 + frac + frac + frac + frac + frac + frac + frac cdots</math>


Since the sum of the reciprocal of every power of two is <math> textstyle 1 = frac + frac + frac + frac + cdots</math>, subtracting the terms with powers of two from x gives

<math>x - 1 = 1 + frac + frac + frac + frac + frac + frac + frac + cdots</math>


Repeat the process with the terms with the powers of three: <math>textstyle frac = frac + frac + frac + frac + cdots</math>

<math>x - 1 -......
...

Read More

No feeds found

All
Posting your question. Please wait!...


No updates available.
No messages found
Suggested Pages
Tell your friends >
about this page
 Create a new Page
for companies, colleges, celebrities or anything you like.Get updates on MyPage.
Create a new Page
 Find your friends
  Find friends on MyPage from