Factorizer2. 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:

produces (by clicking on

Compute) the following output (in about a second):

When

Show number of prime factorsis checked we get:

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).

produces the following output:

produces the following output:

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

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

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

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.

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):

- The number of primes in the range 1,000,000 through 2,000,000 (namely, 70,435) was calculated in 26 seconds.
- The 1,409 primes in the range 2,000,000,000 through 2,000,030,000 were calculated and displayed in 37 seconds.
- The prime decompositions of all 501 numbers in the range 1,000,000 through 1,000,500 (with 39 primes found) were calculated and displayed in 23 seconds.
- The proper factors of the 101 numbers in the range 2,000,000,000 through 2,000,000,100 were calculated and displayed in 31 seconds (with 6 primes found).
- The 862 proper factors of 1,914,412,500, which has prime decomposition 2
^{2}.3^{2}.5^{5}.7.11.13.17, were calculated and displayed in ascending order in a few seconds.If a calculation goes on for too long then it can be stopped by clicking on a

Stopbutton:

## 3. Operation With Filter Condition

A filter condition restricts the numbers considered by the program to positive integers in a particular sequence, e.g., 10^{n}+ 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, ... a ^{n}+ b2 1 2, 3, 5, 9, 17, ... a ^{n}- b3 2 3, 5, 11, 29, 83, ... a ^{n}± b3 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 2

^{n}-1) are green.

All Fermat primes (i.e., primes of the form 2

^{n}+1) other than 3 are red.

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