856 Pages

# Blueeighthnote

## aka B.E.N. (Joe)

Admin Maker PNW Employee
• I live in 5-line musical staff
• My occupation is New Graduate Student
• I am A musical note

| Message Wall:Blueeighthnote

Hi, this is Blueeighthnote. Welcome to Prime Numbers Wiki!

I love numbers since I was little. I love prime numbers even more because they are mysterious!

This account has one currently running computer that is contributing to GIMPS by testing ridiculously large Mersenne Primes. There was one former that has retired.

There are new assignments running and I'll be updating once a week or so.

Current assignments: (Last updated on 2017/10/12 17:45 (UTC +8)

Date Received Completion Number Task % Completed Results GPU credit
2017-10-01 2018 (Expected) 282100567 LL First Time Test 0.55% This will take a long while ...
2017-06-16 2017-10-01 245427483 Lucas–Lehmer Primality test Double Check 100.00% Residue: ED904DE726D4623C 72.9127
2017-06-16 2017-07-18 245133237 Lucas–Lehmer Primality test Double Check 100.00% Residue: 997ED9931DF7A5B3. Successfully verified result. 72.4404
2016-07-18 21289271197-1 Trial Factoring up to 2^68 100.00% Found 1 factor: 277221855447310728359 0
2016-06-25 2016-08-01 243,347,251-1 Lucas–Lehmer Primality test Double Check 100.00% Residue: FAF4D2CE1F8C98C3. Successfully verified result. approx. 71

'Non-LL Results

ECM Curves: M13526297 - 6 curves, B1=50000, B2=5000000
ECM Curves: M2671 - 12 curves, B1=110000000, B2=11000000000
no factor for M150830437 from 2^71 to 2^72 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150875807 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150875839 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150875887 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150876109 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150876149 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150876239 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150876263 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M150876337 from 2^69 to 2^70 [mfaktc 0.21 barrett76_mul32_gs]
M154451107: no factor from 2^68 to 2^69
M154451081: no factor from 2^68 to 2^69
M154451009: no factor from 2^68 to 2^69
M154450951: no factor from 2^68 to 2^69
no factor for M177620977 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M177621001 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M177621091 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M177621109 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M188300113 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M188325947 from 2^70 to 2^71 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M228979483 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M228979523 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M228979529 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M228979577 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M228979591 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M228979721 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M228979727 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759749 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759807 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759809 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759827 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759879 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759887 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759903 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759923 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M476759981 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M567676727 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M567676751 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M567676789 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M567676843 from 2^67 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M567676847 from 2^67 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M667676767 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M667676827 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M667676833 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M667676917 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683675869 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683675933 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683675977 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683676061 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683676167 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683676313 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683676391 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M683676397 from 2^67 to 2^68 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M871704341 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M871704391 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M871704433 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M871704451 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M871704583 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]
no factor for M871704601 from 2^68 to 2^69 [mfaktc 0.21 barrett76_mul32_gs]

What is Lucas-Lehmer Primality test, you may ask?!

Well, first, define a sequence of which the next number is the previous number squared minus 2. It is as following:

Define a sequence $\{s_i\}$ for all i ≥ 0 by

$s_i= \begin{cases} 4 & \text{if }i=0; \\ s_{i-1}^2-2 & \text{otherwise.} \end{cases}$

The first few terms are 4, 14, 194, 37634, 1416317954, 2005956546822746114, 4023861667741036022825635656102100994......

Then Mp is prime if and only if

$s_{p-2} \equiv 0 \pmod{M_p}.$

This method was used by G.I.M.P.S. to test if the number is a prime for extremely large numbers. Before apply the LL test, though, it will first check it through trial factoring (up to at least 265) and P-1 factoring (to test if it probably is a prime), since an LL test takes about a month to complete.

I used to be working on creating Prime Number pages, while also adding some information, like the Relationships with other odd numbers.

Even if I know this wiki may not get a lot of views (the subject of prime numbers is pretty cold), I'll still try to count them from 2 to 10,007.

There are a lot of info yet to be added, like properties of a certain number, what type of prime it is, etc, but they will be added eventually.

But first, here's something for fun! Fun prime number: 809809809809893. It is a prime.

This prime tester allows up to 15 digits.

## Divisibility Rules

These are the templates that may be used in the future. All of these text can be hovered and can provide an insight of how to check the divisibility to a certain number.

### Default values

Divisibility of 1Divisibility of 1:
Every number is divisible by 1.
Example: 375 is divisible by 1.
;

Divisibility of 2Divisibility of 2:
All even numbers are divisible by 2, which are
numbers with 0, 2, 4, 6, or 8 as the last digit.
Example: 2,624 is divisible by 2. Its last digit is 4.
;

Divisibility of 3Divisibility of 3:
Sum the digits. If the result is divisible by 3,
then the number is divisible by 3.
Example: For the number 3,177, the sum
of all digits is 18. The result is divisible by 3;
therefore, 3,177 is also divisible by 3.
;

Divisibility of 4Divisibility of 4:
Check the last two digits.
Example: 3,756 is divisible by 4. The last two digits
of the number is 56, which is divisible by 4.
;

Divisibility of 5Divisibility of 5:
Check the last digit. Numbers with 0 or 5 as the last digit are divisible by 5.
Example: 35,515 is divisible by 5. Its last digit is 5.
;

Divisibility of 6Divisibility of 6:
Perform checks of divisibility of 2 and 3. If both
tests are positve, the number is divisible by 6.
Example: For the number 41,556, it is an
even number (last digit is 6), and the sum
of all digits is 21. The result is divisible by 3;
therefore, 41,556 is also divisible by 3.
;

Divisibility of 7Divisibility of 7:
Take away the last digit of the number, multiply it by 2, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 172,781,
1. Take the last digit of the number and multiply it by 2. (1 x 2 = 2)
2. Subtract that number from the rest of the digits. (17278 - 2 = 17276)
3. Repeat the process. (1727 - 6 x 2 = 1715)
4. Continue until the results can be determined. (171 - 5 x 2 = 161)
5. The result is should be divisible by 7. (16 - 1 x 2 = 14)
;

Divisibility of 8Divisibility of 8:
Check the last three digits. Alternatively, if the third
digit is even, directly examine the last 2 digits. If it is
odd, add 4 to the last 2 digits and examine.
Example: 2,168 is divisible by 4. The last three
digits of the number is 168, which is divisible by 8.
;

Divisibility of 9Divisibility of 9:
Sum the digits. If the result is divisible by 9,
then the number is divisible by 9.
Example: For the number 26,613, the sum
of all digits is 18. The result is divisible by 9;
therefore, 26,613 is also divisible by 9.
;

Divisibility of 10Divisibility of 10:
The last digit is 0.
Example: 7,725,610 is divisible by 10. Its last digit is 0.
;

Divisibility of 11Divisibility of 11:
Sum up the odd digits, then the even digits. Subtract
them. Then check if the number is a multiple of 11.
Example: For the number 38,621, the sum of all odd
digits is 10, and the sum of all even digits is 10. After
subtracting them, the result is 0, which is divisible by
11; therefore, 38,621 is also divisible by 11.
;

Divisibility of 12Divisibility of 12:
Perform checks of divisibility of 4 and 3. If both
tests are positve, the number is divisible by 12.
Example: For the number 857,676, the sum of
all digits is 39, divisible by 3; the last two digits
of the number is 76, which is divisible by 4.
;

Divisibility of 13Divisibility of 13:
Similar to divisibility of 7, but this time, take away the last digit of the number,
multiply it by 4, and add it to the remaining digits. Then repeat the process.
Example: For the number 1,864,369,
1. Take the last digit of the number and multiply it by 4. (9 x 4 = 36)
2. Add that number to the rest of the digits. (186436 + 36 = 186472)
3. Repeat the process. (18647 + 2 x 4 = 18655)

4. Continue until the results can be determined. (1865 + 5 x 4 = 1885)
5. The result should be divisible by 13. (188 + 5 x 4 = 208)
6. (20 + 8 x 4 = 52)

Alternatively, the alternating sum method can be used instead. First, group
the numbers of three from right to left. Then alternate between subtraction
and addition. Then apply the above method to determine its divisibility of 13.
Example: For number 107,359,993,
1. The alternating sum is 993 - 359 + 107 = 741
2. Examine the result. Use the 1st method again to reduce the digits if needed.
;

Divisibility of 14Divisibility of 14:
Perform checks of divisibility of 2 and 7. If both
tests are positve, the number is divisible by 14.
;

Divisibility of 15Divisibility of 15:
Perform checks of divisibility of 3 and 5. If both
tests are positve, the number is divisible by 15.
;

Divisibility of 16Divisibility of 16:
If the thousands digit is even, examine if the last
three digits of the number is divisible by 16.
If the thousands digit is odd, add 8 to the number
first, and then examine the last three digits of it.
;

Divisibility of 17Divisibility of 17:
Take away the last digit of the number, multiply it by 5, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 238,493,
1. Take the last digit of the number and multiply it by 5. (3 x 5 = 15)
2. Subtract that number from the rest of the digits. (23849 - 15 = 23834)
3. Repeat the process. (2383 - 4 x 5 = 2363)

4. Continue until the results can be determined. (236 - 3 x 5 = 221)
5. The result is should be divisible by 17. (22 - 1 x 5 = 17)
;

Divisibility of 18Divisibility of 18:
Perform checks of divisibility of 2 and 9. If both
tests are positve, the number is divisible by 18.
;

Divisibility of 19Divisibility of 19:
Take away the last digit of the number, multiply it by 2, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 977,759,
1. Take the last digit of the number and multiply it by 2. (9 x 2 = 18)
2. Add that number to the rest of the digits. (97775 + 18 = 97793)
3. Repeat the process. (9779 + 3 x 2 = 9785)

4. Continue until the results can be determined. (978 + 5 x 2 = 988)
5. The result is should be divisible by 19. (98 + 8 x 2 = 114)
;

Divisibility of 20Divisibility of 20:
The number formed by the last two digits is divisible by 20.
;

Divisibility of 21Divisibility of 21:
Perform checks of divisibility of 3 and 7. If both
tests are positve, the number is divisible by 21.
;

Divisibility of 22Divisibility of 22:
Perform checks of divisibility of 2 and 11. If both
tests are positve, the number is divisible by 22.
;

Divisibility of 23Divisibility of 23:
Take away the last digit of the number, multiply it by 7, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 953,741,
1. Take the last digit of the number and multiply it by 7. (1 x 7 = 7)
2. Add that number to the rest of the digits. (95374 + 7 = 95381)
3. Repeat the process. (9538 + 1 x 7 = 9545)

4. Continue until the results can be determined. (954 + 5 x 7 = 989)
5. The result is should be divisible by 23. (98 + 9 x 7 = 161)
;

Divisibility of 24Divisibility of 24:
It is most efficient to check the divisibility of
24 by checking its divisibility by 3 and 8.
;

Divisibility of 25Divisibility of 25:
Check the last two digits. They should end in
00, 25, 50, 75, which are divisible by 25.
Example: 1,435,675 is divisible by 25 due to
the last two digits of the number being 75.
;

Divisibility of 26Divisibility of 26:
Perform checks of divisibility of 2 and 13. If both
tests are positive, the number is divisible by 26.
;

Divisibility of 27Divisibility of 27:
Sum the digits in blocks of three and see if the result is divisible by 27.
Example: For number 8,435,529,
1. The digit sum in groups of 3 is 529 + 435 + 8 = 972
2. Examine the result.
If it cannot be determined, use the right trim method:
3. Multiply the last digit by 8 and subtract it from the rest. (97 - 2 x 8 = 81)
;

Divisibility of 28Divisibility of 28:
Perform checks of divisibility of 4 and 7. If both
tests are positive, the number is divisible by 28.
;

Divisibility of 29Divisibility of 29:
Take away the last digit of the number, multiply it by 3, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 120,437,
1. Take the last digit of the number and multiply it by 2. (7 x 3 = 21)
2. Add that number to the rest of the digits. (12043 + 21 = 12064)
3. Repeat the process. (1206 + 4 x 3 = 1218)

4. Continue until the results can be determined. (121 + 8 x 3 = 145)
5. The result is should be divisible by 29. (14 + 5 x 3 = 29)
;

Divisibility of 30Divisibility of 30:
Perform checks of divisibility of 3 and 10. If both
tests are positive, the number is divisible by 30.
;

Divisibility of 31Divisibility of 31:
Take away the last digit of the number, multiply it by 3, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 707,699,
1. Take the last digit of the number and multiply it by 3. (9 x 3 = 27)
2. Subtract that number from the rest of the digits. (70769 - 27 = 70742)
3. Repeat the process. (7074 - 2 x 3 = 7068)

4. Continue until the results can be determined. (706 - 8 x 3 = 682)
5. The result is should be divisible by 31. (68 - 2 x 3 = 62)
;

### Custom example values

Divisibility of 3 (17,511)Divisibility of 3:
Sum the digits. If the result is divisible by 3,
then the number is divisible by 3.
Example: For the number 17,511, the sum
of all digits is 15. The result is divisible by 3;
therefore, 17,511 is also divisible by 3.
;

Divisibility of 7 (1,561,049; 41,485,751,279)Divisibility of 7:
Take away the last digit of the number, multiply it by 2, and subtract it from
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 1,561,049,
1. Take the last digit of the number and multiply it by 2. (9 x 2 = 18)
2. Subtract that number from the rest of the digits. (156104 - 18 = 156086)
3. Repeat the process. (15608 - 6 x 2 = 15596)
4. Continue until the results can be determined. (1559 - 6 x 2 = 1547)
5. The result is should be divisible by 7. (154 - 7 x 2 = 140)

Divisibility of 19 (123,177; 88,813,258; 248,334,006,017)Divisibility of 19:
Take away the last digit of the number, multiply it by 2, and add it to
the remaining digits. Repeat the process until the result can be determined.
Example: For the number 123,177,
1. Take the last digit of the number and multiply it by 2. (7 x 2 = 14)
2. Add that number to the rest of the digits. (12317 + 14 = 12331)
3. Repeat the process. (1233 + 1 x 2 = 1235)

4. Continue until the results can be determined. (123 + 5 x 2 = 133)
5. The result is should be divisible by 19. (13 + 3 x 2 = 19)

### Checktrim

There is a powerful template that I just finished that can check how to perform divisibility check on any number N.

The formula is as follows:

1. Take away the last X digits from the number.
2. Multiply it by A (can be positive or negative).
3. Add it to the rest of the digits. (This is due to a (certain number x A) modulo N = 1)
4. The value A can be found in the Checktrim table. It has been organized.
5. Repeat the above steps.

For example, is 813,527 divisible by 7? Using the trim-1-digit method:

1. Take away the last digit 7. Referring the table, we should multiply it by -2 (or 5). In this case, multiplying by -2 seems easier.
2. Add the number to the rest of the digits. 81352 + 7 x (-2) = 81338
3. Repeating the steps. 8133 + 8 x (-2) = 797
4. Repeating the steps again. 79 + 7 x (-2) = 65
5. The result can be seen - Not divisible by 7.

Second example, is 319,969 divisible by 13?

1. Take away the last digit 9. Referring the table, we should multiply it by 4 (or -9). In this case, multiplying by 4 seems easier.
(You may now notice that the sum of both number's individual absolute value is 13, the number we are checking divisible by.
2. Add the number to the rest of the digits. 31996 + 9 x 4 = 32032
3. Repeating the steps. 3203 + 2 x 4 = 3211
4. Repeating the steps again. 321 + 1 x 4 = 325
5. One more time! 32 + 5 x 4 = 52
6. The result can be seen - Divisible by 13.

For the trim table, an automated one has been parsed by the wiki: User:Blueeighthnote/Checktrim

These are a bunch of number that can be trimmed using the trim method. There is a formula to this. (Warning: Mind the long loading times!!!)

## For Relationships of the prime numbers

All numbers should list two numbers before and after individually. For numbers greater than 2,000, only one is required to be provided at the time being.

For numbers below 100, all odd numbers must be provided a reason why it cannot be considered a prime number. For even numbers, it can be skipped.

For numbers between 101 and 1,999, numbers that can be divided by 3 or 5 can be skipped freely. Numbers that can be divided by 7, however, needs to be listed. If a number can only be divided by a number larger than 13, the factorization should be displayed. The limit gets looser as the other factor gets larger. If larger than 101, then only one factor needs to be displayed.

For numbers after 2,000, numbers that can be divided by 7 can be skipped, too. Twin primes, cousin primes, and sexy primes may not have to be listed at the time being.

For (planned) numbers after 4,000, numbers that can be divided by 11 should be skipped. Numbers that can be divided by 13, 17, 19, however, should not be skipped as they are more complicated to calculate by hand.

If we ever get numbers past 10,000, numbers that can be divided by 13, 17, 19 may get skipped eventually, but we then need a template (or even higher-grade javascript writing) to work around the confusions. Factorizations with one factor larger than the other by 3 times can be skipped and just list down one factor.

If we ever get numbers past 50,000, numbers that can be divided by 23, 29 may get skipped eventually, but we need a very strong page telling users how to calculate divisibility by 10 or above.

## Random Prime number Link

Link to a random prime: (Hooray for random pages!)

If you see any number, single or groups on a page, containing a number that is NOT a prime (aside from those listed in Relationships with other odd numbers), please notify an admin for correction. I often make editing mistakes.

A file. Currently up to 3,000, showing prime numbers in a table, displayed as 10n + [1, 3, 7, or 9]:

Another file. Currently up to 2,880, showing prime numbers in a table, displayed as 30n + [1, 7, 11, 13, 17, 19, 23, or 29]:

For verification of the primes on this wiki, here's a list of prime numbers (from other websites):

Or just use the above prime number checker to check.

## My favorite pages

• (281)?
• (283)!?
• 37 (It's my favorite number!)

## Future

Warning, a lot of weird red font inside! --> User:Blueeighthnote/Sandbox

Classes of Prime Numbers. First of some are from Wikipedia, and then we have to come up with original ones.

......

Probable Primes might get included since they are somehow similar to primes, but not primes.

Template testing area: (Template:Plainnumber) 3664

## Others Wikis that I work on

(Status updated on Feb 23, 2018)

Wikis that I have frequently visited:
Wiki Site User
Page
Editing
Status
Notes
Candy Crush Saga Wiki [1] Former-admin, temporary inactivity, expected return in March 2018
1. First Wiki ever edited on.
2. Popular Wiki, often reaches top 100!
3. Though admittedly I'm pretty behind right now since I'm not playing it very often anymore.
Tower of Saviors Wiki (Chinese) [2] Admin, active Data management and error check
Tower of Saviors Wiki (English) [3] User, active In-game content analysis and stuff
Wikis I've been on
Piano Tiles Wiki [4] Bureaucrat, inactive
1. Editing Pages about the game Piano Tiles and Piano Tiles 2.
2. How irresponsible, I should return editing! I will hop in and re-edit the mess I left in March 2018
Candy Crush Soda Saga Wiki [5] Admin, inactive
1. An excellent Wiki for Candy Crush Soda!
2. Unfortunately inactive as my progress is low.
Diamond Digger Saga Wiki [6] Admin, inactive Diamond Digger Saga. Used to have designed a lot of templates for this wiki to use.
Prime Numbers Wiki [7] Bureaucrat, inactive Retired. While this Wiki is a lot of fun, I've lost interest at this point. I may return someday, though.

Community content is available under CC-BY-SA unless otherwise noted.