:flex:
[QUOTE=Kees]Ramsey[/QUOTE]

When m,n, are 5,5 then R(m,n,) is [43,49]

67
Mally :coffee:

at last - one i recognise
 

Mersenne claimed in his famous list that 2^67-1 was prime when it is actually composite.

67 is also the third irregular prime. My maths knowledge is such that i have only a very dim understanding of what that means, let alone why it is important ...

OK: 133, i don't think we've had this one before.


Richard

QUOTE=Richard Cameron]Mersenne claimed in his famous list that 2^67-1 was prime when it is actually composite.

67 is also the third irregular prime. My maths knowledge is such that i have only a very dim understanding of what that means, let alone why it is important ...

OK: 133, i don't think we've had this one before.


Richard[/QUOTE]
Thanks Richard.
Irregular primes? I went on a wild goose chase and ended up with Bernoulli numbers.
Evidently 67 divides B34 (I think) which is a monster of a number and best left alone.
I meant something simpler and give a Hint.
67 gives Palindromes in different bases.<10.

133 is the smallest number n for which the sum of the proper divisors of n divides phi(n). phi (n) is the Totient Function (Math World)

This is an interesting one.

3.3598856.... ( A constant worth recognising)

Mally :coffee:

[quote=Richard Cameron]67 is also the third irregular prime. My maths knowledge is such that i have only a very dim understanding of what that means, let alone why it is important ..[/quote]
For many of us, the importance of irregular primes is that for a while in the mid-19th century, Fermat's Last Theorem had been proven for all "regular" prime exponents, but not for the "irregular" ones such as 67.

[URL="http://mathworld.wolfram.com/IrregularPrime.html"]http://mathworld.wolfram.com/IrregularPrime.html[/URL]

[quote]Mersenne claimed in his famous list that 2^67-1 was prime when it is actually composite.

< snip >

OK: 133, i don't think we've had this one before.[/quote]
Just replace the "^" in "2^67-1" with "*" to produce 133 ...

[quote=mfgoode]3.3598856.... ( A constant worth recognising)
[/quote]
The title of this thread is "Special [I]whole[/I] numbers..." :smile:

Special whole numbers
 

[QUOTE=cheesehead]For many of us, the importance of irregular primes is that for a while in the mid-19th century, Fermat's Last Theorem had been proven for all "regular" prime exponents, but not for the "irregular" ones such as 67.
[URL="http://mathworld.wolfram.com/IrregularPrime.html"]http://mathworld.wolfram.com/IrregularPrime.html[/URL]
Just replace the "^" in "2^67-1" with "*" to produce 133 ...

The title of this thread is "Special [I]whole[/I] numbers..." :smile:[/QUOTE]

:bow: Thank you cheesehead for the short cut to 67. Yes I visited the site you have given ('wild goose chase').

Besides the FLT however, you have not given the reason why and how these irregular primes are generated, satisfying the Bernouli criterion viz: it should divide the numerator of a bernoulli number. However there is no necessity for this ('well left alone')

The known irregulars are 37, 59, 67, 101, 103, 149,157. There should be one more as I remember distinctly reading 8 'B' numbers and Im quoting from memory.

Regards 'Special Whole numbers' if you have diligently followed this thread you will find that on several instances (posts) this rule was broken and no one objected and so it carried on.

Why the objections now?

May I stress the point that my aim is to diseminate knowledge to others and not tom foolery, the kind to send people 'to monkey up the tree' ( a game I played as a kid)!

To waste a persons time on a useless search with no significance is a sin even if one does not believe in sin, call it what you like!
:popcorn: :popcorn:
The constant I have given should at least be heard off if not remembered for all those in the quest of a 100K prize !
Mally :coffee:

R(5,5) is indeed bounded by 43 and 49 as for now. I am a little bit surprised that this is not sharpened because this result is already 10 years old.

As to your constant, Mally, there may have been other post with non-integer numbers but does that mean that you have to do so yourself ?

I think the proper place for the constant would be a thread called 'special numbers'.

Giving you a chance to start such a thread and posting another speciale whole number,

Kees :cat:

R(m,n) is the smallest number of vertices in a graph such that there is either a complete subgraph of order m or an independent set of order n.
Obviously this number must be unique. However for R(5,5) we only know the interval in which it must lie: ([43,49] int N).
The numbers are called Ramsey numbers

Whole numbers
 

[QUOTE=Kees]R(5,5) is indeed bounded by 43 and 49 as for now. I am a little bit surprised that this is not sharpened because this result is already 10 years old.

[Mally: The Pythagorean theorem has not been 'sharpened' for the last 3000 years]

As to your constant, Mally, there may have been other post with non-integer numbers but does that mean that you have to do so yourself ?

[Mally: I am a non conformist as my reply in this fashion illustrates and suits me]

I think the proper place for the constant would be a thread called 'special numbers'
.
[I agree]

Giving you a chance to start such a thread and posting another speciale whole number,

[Thank you] :love:

Kees :cat:[/QUOTE]

47

Mally :coffee:

Many answers are possible to 47, but I think for this forum this must be by far the prettiest:
17296, 18416 are amicable numbers (like perfect numbers, but with 2-period repetition in stead of 1-period)

Thabit ibn Qurra derived a formula to generate some of these numbers:

p = 3 × 2^(n-1) - 1,
q = 3 × 2^n - 1,
r = 9 × 2^(2n-1) - 1

If all three are prime, then A=2^n*p*q and B=2^n*r are amicable numbers.
Dotting in n=4 we get:

p = 23
q = 47
r = 1151

all prime, and A=17296, B=18416

:cat:

Forgot to post a "special number", so here goes

672

amicable numbers.
 

[QUOTE=Kees]Many answers are possible to 47, but I think for this forum this must be by far the prettiest:
17296, 18416 are amicable numbers (like perfect numbers, but with 2-period repetition in stead of 1-period)

Thabit ibn Qurra derived a formula to generate some of these numbers:

p = 3 × 2^(n-1) - 1,
q = 3 × 2^n - 1,
r = 9 × 2^(2n-1) - 1

If all three are prime, then A=2^n*p*q and B=2^n*r are amicable numbers.
Dotting in n=4 we get:

p = 23
q = 47
r = 1151

all prime, and A=17296, B=18416

:cat:[/QUOTE]
:bow: Hats off to you Kees you are a genius!
You have found a needle in a haystack. Excellent work and I did some reading up and find the formula given by Thabit ibn Quarra is very lucidly explained by you( Much easier than Math World)
I believe it was re-discovered by Fermat and then by Descartes who gave their own contribution of Amicable numbers with much larger ones.

But the Italian Paganini just 16 yrs old stunned the math world in the !8th century by giving the second smallest A/No.s viz. 1184 and 1210 which escaped all the rest in their tables of A/No.s. Milli Gratia! to the italians.
Till date there are more than 10 million pairs known.

However my small number 47 has to do with the Fib. and Lucas series. I will give it in my next post if not answered by then.
Mally :coffee: