# Talk:Lucky number

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Field:  Number theory

## Untitled

The entire article (including the References) has been copied word-for-word from the following link - AbsoulteAstronomy.com. I am not sure if it violates the copyrights or not. --Bhuvan 17:15, 11 May 2005 (UTC)

Go check out the bottom of the page you linked there. Notice removed. --SomeoneWhoCanRead

## Infinite Primes

Article states at the end: "There are infinitely many lucky numbers. It is not known whether there are also infinitely many lucky primes..."

Haven't done maths for 20 years but there must be an infinite number of lucky primes, isn't that right?

82.213.248.138 20:06, 6 November 2005 (UTC)El Choco

No, we actually can't tell from that statement. For example, we could also say: "There are infinitely many even numbers. It is not known whether there are also infinitely many even primes" (where I have simply replaced "lucky" with "even"). My statement is clearly not true, since there is only one even prime. The two categories are almost completely disjoint. Just because there are an infinite number of prime numbers and an infinite number of lucky numbers doesn't mean that there are an infinite number of lucky primes. N Shar 02:13, 13 October 2006 (UTC)

## Poorly-explained algorithm

It seems like this could be described more clearly. As it is, the even-number-removing step seems out of place. Something like this, maybe?

``` for n=2 to infinity, do
let x = the nth number of the current list;
remove every xth number from the current list;
```

Rob* 07:00, 5 April 2006 (UTC)

I agree that the "1st step" does not fit well into the general scheme (among others, because 2 itself is also removed, and because it's the 1st step but the 2nd number in the list, while the 2nd step uses again the 2nd number from the list). On mathworld, EWW starts out with the odd numbers (which of course could also be considered as odd...)
An alternative description could be: start with the smallest ("remaining") number > 1, which is 2. Then use the smallest remaining number > 2, which is 3. Then, the smallest remaining number > 3, which is 7; etc. — MFH:Talk 17:04, 2 June 2009 (UTC)
PS: This would amount to the following algorithm:
```  L = all positive integers; X = 1 (= min(L))
do forever
X = min { y in L | y > X }
remove every X-th number from L
end do
```
OTOH, it seems that a function islucky(n) is more difficult to write than isprime(n) (which needs only checking for divisibility by 2 and then by all odd numbers >= 3 up to sqrt(n)).
Is there any hint somewhere about how this could be done "somehow efficiently" (in particular, limiting memory usage)? — MFH:Talk 17:42, 2 June 2009 (UTC)

I was just noticing this myself. There's an ambiguity in the way it's described. Is the interval by which to count at each pass:

1. the next survivor, after the one that was used on the last pass?
2. the 2nd, 3rd, 4th, 5th, etc. of those left at each stage?
3. looked up using the previous interval as an index into the list?

Of course, 1 and 2 are equivalent, since after pass 1 the interval (hence the index of the first number to be eliminated in the pass) is always greater than its index in the list. And 3 doesn't produce the sequence given in the article. But still, it would be better to phrase it in a way that eliminates this ambiguity. Here's an idea: changing "The third surviving number" to "The next surviving number" will avoid the temptation to think it's third because the previous interval was 3, and therefore that the 7th surviving number will be picked as the interval for the next pass. (As would showing one more pass of the process, for that matter.)

But maybe there is indeed a better way to describe it that avoids the incongruity of the first step differing from the remainder. MFH's description seems to have been tweaked from that on MathWorld. But does anybody here have access to the original description by Gardiner et al? — Smjg (talk) 20:16, 24 April 2012 (UTC)

## Lack of citation

This claim is easy to find verbatim on the internet, but I can't find any more information, does somebody have a more precise year, or possibly details about the story? "Stanisław Ulam was the first to discuss these numbers, around 1955. He named them "lucky" because of a connection with a story told by the historian Josephus." — Preceding unsigned comment added by 68.149.25.30 (talkcontribs) 05:42, 24 October 2006‎

## "Largeness" of lucky numbers

Do lucky numbers form a large set (just like primes)? If not, is there known any upper bound of sum of reciprocals? — Preceding unsigned comment added by Wojowu (talkcontribs) 13:12, 21 August 2012 (UTC)

The article says: "Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem". This is confirmed in one of the external links http://mathworld.wolfram.com/LuckyNumber.html which says: "Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is 1/ln N, just as the prime number theorem". So the sum of reciprocals diverges like the prime numbers. oeis:A000959 has more references about lucky numbers. PrimeHunter (talk) 13:27, 21 August 2012 (UTC)

## Ulam spiral analog

File:/tmp/luckySpiral.png
The lucky spiral

I produced, for lucky numbers, a picture analogous to the Ulam spiral. It appeared to show similar diagonal stripes. However, a larger spiral with lucky numbers to 200,000 looks considerably more random. I think I will not bother to upload the picture when my account is verified in a few days. David Lambert. Preceding unsigned comment added by 69.205.204.253 (talk) 20:34, 21 March 2013 (UTC)

## Frequency

Twin lucky numbers and twin primes also appear to occur with similar frequency.

What is the precise meaning (and source) of this sentence?

Anne Bauval (talk) 06:54, 3 September 2014 (UTC)

## "the sieve of Josephus Flavius"

Gardiner, Lazarus, Metropolis and Ulam did not suggest to call the defining sieve "the sieve of Josephus Flavius". They suggested this name for another sieve. — Preceding unsigned comment added by 134.147.5.237 (talk) 15:36, 24 March 2015 (UTC)