Factorizer
2. Operation Without Filter Condition
3. Operation With Filter Condition

2. Operation Without Filter Condition

When no filter condition is set the program examines all numbers in the specified range from the start number to the end number.

For example, the following input:

1234567890 to 1234567905 input

produces (by clicking on Compute) the following output (in about a second):

1234567890 to 1234567905 output


When Show number of prime factors is checked we get:

With 'Show number of prime factors'

Of the two numbers in parentheses, e.g. (5, 1.123), the first is the number of prime factors (in this example there are five: 2, 3, 5, 3,607 and 3,803). The second is the EK value (defined in Section 7).


The following input:

987654321 to 987654530 input

produces the following output:

987654321 to 987654530 output


The following input:

produces the following output:


Requesting prime pairs only on the same range of numbers produces the following output:

987654321 to 987654530 output

The largest prime pair which can be discovered using this program is 2,147,482,949 and 2,147,482,951.  It is not known whether there are an infinite number of prime pairs, but this is conjectured to be so.


Factorizer allows you to find all proper factors of a number (i.e., all factors other than 1 and the number itself). E.g.:

factors of 12,367,296

You can get all proper factors of a range of numbers, e.g.:


Copy output to clipboard A lengthy output can be copied to the clipboard and then pasted into a word processor such as Notepad or Wordpad (from which it can be printed or saved to disk).

The text in the output window is editable, so you can delete parts of it or add comments as needed.

Save/restore state The state of the program can be saved at any time, allowing you to explore another line of thought and then return to the saved state by restoring it.

The program saves its state when you quit and restores it when it starts up again.


As regards speed, here are some timing results (all from runs on a 600 MHz Pentium III PC):

If a calculation goes on for too long then it can be stopped by clicking on a Stop button:



3. Operation With Filter Condition

A filter condition restricts the numbers considered by the program to positive integers in a particular sequence, e.g., 10n + 2, which (for n >= 0) consists of the numbers 3, 12, 102, 1002, ...

There are nine possible types of filter condition (division must be exact):

Filter type Examples
  a b n an integer and resulting number >= 1
a*n + b 3 1 1, 4, 7, ...
a*n - b 7 12 2, 9, 16, ...
a*n ± b 7 12 2, 5, 9, 12, 16, 19, ...
a/n + b 1000 1 2, 3, 5, 6, 9, 11, 21, 26, ...
a/n - b 105 1 2, 4, 6, 14, 20, ...
a/n ± b 105 1 2, 4, 6, 8, 14, 16, 20, 22, ...
an + b 2 1 2, 3, 5, 9, 17, ...
an - b 3 2 3, 5, 11, 29, 83, ...
an ± b 3 2 1, 3, 7, 5, 11, 25, 29, 79, 83, ...

Examples of results obtained with various filter conditions in effect are given below.

It looks as if all colors of the form 12*N + 1 are black, but 385 =12*32 + 1 is white.

All Mersenne primes (i.e., primes of the form 2n-1) are green.

All Fermat primes (i.e., primes of the form 2n+1) other than 3 are red.

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