 |
PRIME NUMBERS
Primes vary according to the number base
Primes are independent of the number base (invariant to use a mathematical term). After all, computers use binary or hexadecimal for their calculations. The bases are just different ways of representing numbers. Having said that, if I represent numbers as I can, by the exponent, as in a logarithm where the log is the power, then clearly this is not the case, because the arithmetic is that of the power rather than of the number itself. With logs, what is a multiplicative relationship in ordinary maths, is instead is represented by addition. ie X times Y =Z log Z = log (X times Y)= log X + log Y
Keith Martin - Prime Numbers
Well done, yet another absorbing program. Are numbers an abstract construction of human thought, or do they have an independent existance?
Prys Lewis - off topic - sorry
Please Please Please, can we have some of the archive as podcasts so that I can make the journey into work bareable 5 days a week instead just the one.
Paul - prime numbers
Why is it surprising that Reinmann found flat spots on the 1/2 line? Doesn't 1/2 have something to do with "even-ness"? Which is the opposite of prime-ness. (except for the number 2)
Dubby Granger - Primes (infinite series)
Maybe a simple example of an "infinite sum" (i.e. taking limits) would help too. Consider 1/2 + 1/4 + 1/8 + 1/16 + .... The nth term is 1/(2^n). With some simple algebra, the sum of the first n terms of the series is very easily shown to be 1 - (1/2^n). For example, the sum of the first four terms is 1 - 1/16 = 15/16. In fact the sums of the first one, two, three ... terms go: 1/2, 3/4, 7/8, 15/16, 31/32 ... and so on. So in this example it's easy to see that "in the limit" (as mathematicians say), i.e. as n "tends towards infinity", the sum of the first n terms "tends towards" 1. So 1 is then defined to be the "sum of the (infinite) series".
Robert... Archive
Please make the full archive available via itunes rather than just the current episode. This would be a superb "On the Go" resource. Who's with me?
Digby Barker - Prime Numbers
Excellent choice of topic & panel. But do we know if the Riemann Hypothesis is decidable or not (cf Godel's Theorem etc)? And a programmme about the latter would be good too !
Robert J Baker, Prime Numbers
I wonder if there's any link between the Riemann hypothesis and an interesting property I found when experimenting with Stirling's Approximation for computing the factorial of any number; I found that when Stirling's Approximation is called upon to give the factorial of a negative number, it only gives a real result for numbers of the form k+0.5 (where k is any negative integer); for other negative numbers it gives an imaginary result. If this isn't connected to the similar property of the Riemann hypothesis given in the programme, it's certainly a fascinating coincidence...
Susan Lane Prime Numbers
Sieve of Eratosthenes Could someone explain. On the programme Jackie Stedall said: if you cross out everything divisible by 2 (even numbers), everything divisible by 3, and everything divisible by 5, what you are left with in the net are the prime numbers. What about 49, 121, etc. (primes squared)? What have I missed?
Natasha Bedford - Prime Numbers: Vox Populi
Prime Time! Absolutely wonderful! My only grouse is, scores of people seem to be writing in, asking for the archives to be made downloadable: but you haven't! Vox Populi, Vox Dei! Could you please make an exception for IOT and make the archives downloadable just for a short period (a week, perhaps) ? I'd also love to get in touch with other aficionados of IOT: natasha.smith21@googlemail.com
C Ireson Prime Numbers
Brilliant programme. Could you put the mathematical series/equations on the web site. I was fascinated by the squares and the reciprocals etc but sadly can't remember the details. I'd like to have the details. Thank you
j.hyde: Prime Numbers
This was a fascinating programme! Sadly I couldn't hear all of it as I was in the car and reached my destination. I was particularly interested in the description of a prime number as one which could not be expressed geometrically: what about a circle? Surely this too is confirmation of what was being said about prime numbers being the building blocks? Prime numbers -> circles -> eternity -> ???? At school we were always taught that a prime number was one which was divisible only by itself and one - but that would also make one a prime number which I note that it isn't. I shall have to listen to the rest of the programme now
.
Brian Fiddian Primes
Fantastic programme - well done for tackling this sort of topic. The presenters did a great job of explaining some hard comcepts in simple terms and without a blackboard and chalk. Although it was a bit like going out for a jog with a very good runner - they could not resist stepping on the gas from time to time. More please!
Terry Collins Prime numbers
A response to Miles: Miles points out that it is meaningless to talk about the sum of the reciprocals of the squares of the natural numbers because there are an infinite number of these, and so they can never be summed. This is perfectly correct. But when mathematicians talk about such an infinite "sum" they are really talking about the "limit" of the sequence of finite sums of the series. The concept of "limit" is a more abstract notion than a simple sum, but may be regarded as a generalisation of it, and it allows us to attach a meaning to an infinite "sum". Miles also notes that pi cannot be written down exactly because the decimal representation of it (3.14159...) never terminates. This is true also, but the definition of pi (circumference divided by diameter) IS finite and precise. It is only when we try to write pi in decimal notation that it appears to be imprecise.
Terry Collins Prime numbers
Dickon Povey is concerned that whether a number is prime depends on the number base used to represent it. This is not the case. If we adopt the method mentioned in the program of arranging beans in rectangular formations, then 13 beans cannot be arranged in a rectangle, no matter how we express the number 13. So, the primeness of 13 does not depend on the number base used to represent it.
Guy Hoghton: Prime Numbers
Patricia: What you are really saying is that the recurring part written as a whole number makes a multiple of 9. In fact this holds for any recurring fraction whenever the denominator is not divisible by 3 (not just prime numbers). Try for example 3/14 = 0.2142857142857... The recurring number is 142857 which is a multiple of 9. This comes from the fact that the recurring part can be written as 142857 / 9999990. The denominator clearly has a factor of 9 which must be balanced by a factor of 9 in the numerator because the fraction 3/14 has no common factor with 9 in its denominator.
Tony Moon, Science
What about a programme on Moore's Law, its history and likely future progress, with its implications.
Prime Numbers
I owe a lot of my education to the BBC. The great thing abt IOT is that it seeks to deal with science and the arts, as well as philosophy and history. But the Primes are absolutely fascinating: they're like outer space, a frontier of knowledge that beckons humanity, and which has fascinated us for so long. This really was 'Prime Time'. Well done Melvyn, guests, and BBC.
Imaya: Prime Numbers
A fantastic programme. Bravo to the contributors for discussing a complex topic in the simplest of language which everyone could understand. If only my lectures at University were as interesting and engaging. Enjoyed in thoroughly.
Ian: Prime numbers
I'm a regular listener to IOT; and long may it continue. However I found today's discussion almost totally baffling and abstruse (even more than the programme on quantum physics, which is saying something!). I was soon lost (all that talk about zeta and 'powers of...').Eventually I had to switch over to R3. Judging from the opinions posted here I'm in a very small minority, but I hope programmes dealing with such abstract concepts are kept to a minimum.
Andy Pryke - The Sieve of Eratosthenes
Cross out the two's Cross out the three's The sieve of Eratosthenes Follow this method quite sublime The numbers which remain are prime Andy
Prime Numbers
My question is, do prime numbers only relate to the decimal system. After all if we were to count in the binary system then the observation relating to the life cycle of cicadas and the 17(prime in decimal) years become a moot point.
David Hughes - Prime numbers
Well done - a very good edition of a reliably worthwhile programme. I hope however that the editor will try to stop contributors treating the listeners as complete idiots, as most of your listeners are unlikely to fall into this category (they may be ignorant of the subject but that is entirely different) and it does grate. One example: "I think most people probably know what a graph is" - !
Mark Credland - Prime Numbers
Great show... Again! Just two questions. 1) Do all the facts stated by your guests about prime numbers hold up when working in a number system with a base that is not 10? 2) Would the collapse of the world's financial system, which you alluded to in the introduction, be caused by a lack of secure internet communications or did you run out of time to discuss this interesting idea?
Chris Miller: Prime numbers
Miles: not a silly question at all! When mathematicians say an infinite sum 'converges' to a given answer, they really mean that by taking a sufficiently large number of terms in the series, the sum can be made as near to the 'answer' as desired. The sum of the reciprocals of the squares converges rather slowly - so, if you want pi to a thousand places you might need to take a million terms of the sum, if you want pi to a million places, you might need a trillion terms (there is a formula that allows these values to be calculated). But no matter how accurately you want to calculate pi, there is a (very large) number of terms of the sum that will deliver the required accuracy - therefore the sum 'converges' to pi divided by 6. Hope this helps! (Another brilliant programme, BTW.)
Roger Pickering; Prime Numbers.
Caught the programme by accident while driving. The guests did a wonderful job in making the subject almost understandable to me, a dunce in maths. Melvyn Bragg's interjections were very amusing, re-assuring to the audience and kept me aboard "by my fingernails" Very nearly as entertaining as his interview with Willie Nelson for another TV Channel.......great!!
Peter Thornber - re:PRIME NUMBERS
Surely, working in "base 10" breaks down into working in "base double 5"? What work has been done into the juxtaposition of squares on, or adjacent to, the spine of this system, i. e. 0, 5, . . .? Does the Fibuonacci sequence relate to the prime number sequence - and, if so, how? Does the zeta-function work for odd number powers + or _ 2? . . . 4?
Antony : Prime Numbers
I didn't understand the conclusion drawn about the underground cicadas, their predators and prime numbers. If cicadas were smart enough to work out that they needed to avoid predators by staying underground, then why would they come out at all?
peter20001 Prime Numbers and Reality
Do prime numbers also exist in other number bases, other than base 10 and are their characteristics the same and are they only important to maths and science in the anthropormorphic sense?
Rory O'Brien: Prime Numbers
That so abstract a topic was made so intelligible in so short a time says a lot for the power of fairly ordinary words and for the participants' mastery both of language and of their subject. This is as much a part of the fascination of the programme as the topics themselves. Thanks.
PRIME NUMBERS
Melvyn, thank you for an excellent programme that I found fascinating. Maths and Science is almost utterly ignored by everyone else in the media apart from you. Stephen Roe, Dollar, Scotland
Miles: Prime Numbers
Excuse me if this is a silyl question and I've missed something really obvious. How is it possible to say that the sum of the reciprocals of the squares of natural numbers, is the same as pi over 6? Surely as there are an infinite number of natural numbers, and you can calculate pi to an infinite number of places, they will never equal each other? Thanks Miles
Dickon Povey: Prime Numbers
Dear Melvyn I thoroughly enjoyed this morning's broadcast of "In Our Time" but I have a worrying concern. I am certainly not a theoretical mathematician but I have been in the electronics industry all my life, which of course is mathematically based. My worry is this. All the interesting argument was concerning prime numbers in "base 10" as though this was some natural or god given system. However it is an arbitrary number system of human convenience. Would all the theories expressed by your panel apply equally to all other number systems? Yours with great interest Dickon Povey
Ernie : Prime Numbers
You deserve an award for sheer courage. Complex Mathematics on radio! I did maths at college many years ago but it was hard work and never so well presented. Brilliant!
Gavin: Prime numbers
Thanks to Melyn and the production team for yet another absolutely brilliant programme. The zeta function and the Riemann hypothesis were explained so neatly and clearly, and the conversation held so beautifully together. This programme is radio's finest topical discussion.
Tony : Prime Numbers
An excellent programme, which held my attention from start to finish. I look forward to listening to it again!
|
|
