User:Zfishwiki

I've not done very much so far. Possibly my biggest creation was the article for the Rhodes-Milner Round table group.

Also I've added a few adddendums to the talk page on the unexpected hanging paradox. Recently I've been interested in the single transferable vote, and particularly the Meek's method.

Euler's Sieve

Euler in his Proof of the Euler product formula for the Riemann zeta function came up with a better version of the sieve of eratosthenes in which each number was eliminated exactly once.

This runs as follows. We start with the sequence of natural numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35...

we then multiply each member of this sequence by two and remove from the original sequence to obtain

1   3   5   7   9    11    13    15    17    19    21    23    25    27    29    31    33    35...

we then multiply each member of this sequence by three and remove from the working sequence to obtain

1       5   7        11    13          17    19          23    25          29    31          35...

we then multiply each member of this sequence by five (the first number after one) and remove from the working sequence to obtain

1           7        11    13          17    19          23                29    31            ...

Each member of the working sequence we remove is removed exactly once unlike the original Eratosthenes algorithm. If we continue with this process indefinately we will be left with just the number one in our working sequence as in the Proof of the Euler product formula for the Riemann zeta function.

If however we store the eliminated prime numbers as they arrive in a queue and combine them with the working sequence when we have exceeded the square root of the upper limit of our range, we have the desired sequence of prime numbers. Of course we must also eliminate the remaining number one in our working sequence.

Sandbox

This is a previous addition to Talk:Sieve of Eratosthenes which I have deleted and updated in User:zfishwiki#Euler's sieve

I have come up with a more efficient version of the Sieve of Eratosthenes in which each number is eliminated once and only once. I arrived at it by reading the Proof of the Euler product formula for the Riemann zeta function. I very much doubt if it's an original idea, so it doesn't count as new research.

Here it is -:

[1A](2) 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25...
[1B]       4     6     8     10      12      14      16      18      20      22      24    ...
[2A]   (3)    5     7     9      11      13      15      17      19      21      23      25...
[2B]                      9                      15                      21                ...
[3A]         (5)    7            11      13              17      19              23      25...
[3B]                                                                                     25...
[4A]               (7)           11      13              17      19              23        ...
[...]

The Bracketed terms refer to prime numbers as they are eliminated. We obtain (for example) sequence [2B] by multiplying the bracketed first member of [2A] with each and every member of [2A]. So we get -:

(3)*(3)=9, (3)*5=15, (3)*7=21, (3)*9=27,...

We obtain sequence [3A] by starting of with [2A], removing the bracketed member and deleting all members in [2B], all of whose members actually do occur in [2A]. We then bracket the first member

Please let me hear your feedback as to whether this ought to be included in the article.