Well here’s something you might not be aware of.

An ancient proof of the infinity of the primes goes something like this.  Multiply all the primes together up until you get to what you think is the last prime, and then add 1, and you will generate a further prime.  At first glance this seems to look good.  If we take the first 6 primes which are: 2, 3, 5, 7, 11 & 13 then we can generate primes in the following way.

2 x 3 + 1 = 7 (prime)

2 x 3 x 5 + 1 = 31 (prime)

2 x 3 x 5 x 7 + 1 = 211 (prime)

2 x 3 x 5 x 7 x 11 + 1 = 2311 (prime)

2 x 3 x 5 x 7 x 11 x 13 + 1 = 30031 (not a prime!!)

Bet you didn’t see that one coming 🙂

 

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I let my images speak for themselves.

I do not need to “gloss up” a mediocre image with flowery prose – mainly because I don’t produce any mediocre images 🙂

 

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Mathematics as Language

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Got today’s EPOD with the Coma-Virgo Supercluster of Galaxies panorama.  A 2-framer taken with the Canon 200mm lenses and around 5-hours of 15-minute subs per frame.

Thank you Jim at EPOD for continuing to publish my work 🙂

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If you have need for high-resolution, high quality images for books, magazines, posters, framed artwork, inspirational image for offices/buildings – then take a look at the following fine-art photo-libraries for your next project:

  1.  For scenes of the New Forest:  https://www.flickr.com/photos/12801949@N02/albums/72157661372909775
  2.  For Mathematical structures:  https://www.flickr.com/photos/12801949@N02/albums/72157625498790798
  3.  For coastal scenes:  https://www.flickr.com/photos/12801949@N02/albums/72157625231061479
  4.  For astrophotography:  https://www.flickr.com/photos/12801949@N02/albums/72157622020758946
  5.  For Dartmoor panoramas:  https://www.flickr.com/photos/12801949@N02/albums/72157621877819395
  6.  For Photomicroscopy:  https://www.flickr.com/photos/12801949@N02/albums/72157621951960100
  7.  For Macrophotography:  https://www.flickr.com/photos/12801949@N02/albums/72157621951344990
  8.  For High-Speed Flash Photography:  https://www.flickr.com/photos/12801949@N02/albums/72157621810713655

 

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Could I please re-iterate:  None of my images are available “free of charge”.  Please take a moment to look at the “Image Agency and Copyright Notice” section.

Thank you.

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I have got a LOT of negative comments to make about book Publishers, but here’s the one I want to share with you today.

I was asked to review a book that was being considered for publication – all o.k. so far.  But two questions the publisher wanted answers to were:

  1.  Are there any competing books?
  2.  Potential readership/market for the book?

Now this is a doubly annoying question for the following reasons.  Firstly the Author(s) have already given full answers to these questions on submitting the manuscript.  Secondly – if this information is so important, why don’t the publishing staff do a little bit of work AND LOOK UP THE ANSWERS TO THESE QUESTIONS THEMSELVES???

Rant over.

 

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http://www.simulation-argument.com/simulation.html

 

 

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If you have played with the square root function on an electronic calculator you may have noticed something odd.  It looks like if you take the square root of ANY number that is not actually a perfect square, then you get an irrational number, with the decimal places spilling over the edge of your calculator’s display.  Is this in fact true?  Is the square root of any number that is not itself a perfect square an irrational?  Let’s find out.

We start off in the conventional way by assuming that the square root of 2 IS rational and may be given as m/n where m & n are both integer.

1)  √2 = m/n

2)  m2 = 2 n2

That much we have seen before, but now comes the really clever bit.  Factor m & n into their unique prime factors:

3)  m1 m1 m2 m2 m3 m3 … mr mr = 2 n1 n1 n2 n2 n3 n3 … nr nr

Now here’s the problem.  The prime number 2 will appear an even number of times on the left hand side of the equation (if it appears at all) and an odd number of times on the right.  Since the decomposition into primes is unique the prime number 2 cannot appear an even number of times on one side of the equality and an odd number of times on the other.  So the square root of 2 cannot be written in the form m/n with m & n integer.  The square root of 2 is an irrational number.

A) Although the above was shown to be true for root 2, the same argument of course would hold true for any prime number. The square root of any prime number is an irrational.

B) Replace 2 with any integer that is not a perfect square. Now that integer may be decomposed into its prime factors, and, if the number is not a perfect square, then once again we will have an odd number of primes on one side of the equality and an even number of primes on the other.  The square root of any integer that is not a perfect square is also an irrational.

C) Finally, replace 2 with any perfect square. With even numbers of primes on both sides the solution is trivial.

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Admittedly Cantor was in his writings not very explicit about what he did take the set theoretic universe as a whole to be. One problem is that it is not in every instance clear whether he has a theological or a mathematical conception of absolute infinity in mind. Indeed, he argues that it is the task not of mathematics but of ‘speculative theology’ to investigate what can be humanly known about the absolutely infinite. The following passage, for example, leans heavily to the theological side: I have never assumed a “Genus Supremum” of the actual infinite.  Quite on the contrary I have proved, that there can be no such “Genus Supremum” of the actual infinite. What lies beyond all that is finite and transfinite is not a “Genus”; it is the unique, completely individual unity, in which everything is, which comprises everything, the ‘Absolute’, for human intelligence unfathomable, also that not subject to mathematics, unmeasurable, the “ens simplicissimum”, the “Actus purissimus”.  In this quotation, Cantor speaks of the necessity of ‘knowing’ the domain of variation through a ‘definition’. Surely Cantor is merely sloppy here, and we should discount the epistemological overtones. Another slip can be detected in Cantor’s use of the word ‘set’ in this quotation. Cantor means the argument to be applicable not just just to sets but also to absolute infinities. which is by many called “God”. All this is related to the fact that in an Augustinian vein, Cantor takes all the sets to exist as ideas in the mind of God.

And that last sentence rings a bell too 🙂  For Ramanujan said “An equation for me has no meaning, unless it represents a thought of God”.

We seem to be homing in on something here!

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