Weak prime

Weak primes (Engl. Weakly Prime Numbers or digital Delicate Prime ) are prime numbers , which always lose in case of change of any single digit in any other figure their prime property. As a weak prime numbers (Engl. Weak Prime ) but also for encryption called improper primes.

Examples

• The prime number is a weak prime number because: ${\displaystyle p=294001}$

If you change just one of the six digits, you only get composite numbers, which are therefore not prime numbers:

0 94001, 1 94001, 3 94001, 4 94001, 5 94001, 6 94001, 7 94001, 8 94001, 9 94001

2 0 4001, 2 1 4001, 2 2 4001, 2 3 4001, 2 4 4001, 2 5 4001, 2 6 4001, 2 7 4001, 2 8 4001

29 0 001, 29 1 001, 29 2 001, 29 3 001, 29 5 001, 29 6 001,, 29 7 001, 29 8 001, 29 9 001

294 1 01, 294 2 01, 294 3 01, 294 4 01, 294 5 01, 294 6 01, 294 7 01, 294 8 01, 294 9 01

2940 1 1, 2940 2 1, 2940 3 1, 2940 4 1, 2940 5 1, 2940 6 1, 2940 7 1, 2940 8 1, 2940 9 1

29400 0 , 29400 2 , 29400 3 , 29400 4 , 29400 5 , 29400 6 , 29400 7 , 29400 8 , 29400 9

Overall, in this case one has to examine numbers to see if they are composite numbers. ${\displaystyle 9\cdot 6=54}$

• The first weak prime numbers are:

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, 5152507, 5564453, 5575259, 6173731, 6191371, 67327179, 7412679, 6191371, 62327179, 7412679, 6732769, 7412679789, 67327179, 741269789, 67327179, 7412679789, 67327179, 741269789, 67327179, 74126797 ... (Follow A050249 in OEIS )

• The largest currently known weak prime is (as of December 10, 2018):

${\displaystyle {\frac {17\cdot 10^{1000}-17}{99}}[phone]=17\ldots 1738858369}$

This number has 496 , which are directly behind each other, it has digits and was discovered in March 2007 by Jens Kruse Andersen. ${\displaystyle 17}$ ${\displaystyle 1000}$

properties

• In order to determine whether a -digit prime number is a weak prime number, one must control numbers, whether they are compound or not. Only when all the numbers are put together is the -digit prime number actually a weak prime. (see example above) ${\displaystyle k}$ ${\displaystyle 9\cdot k}$ ${\displaystyle 9k}$ ${\displaystyle k}$
• There are infinitely many weak prime numbers.

Proof: see by Terence Tao from 2011.

Weak prime numbers in other number systems

The above section dealt with weak prime numbers in the decimal system , i.e. the base . ${\displaystyle b=10}$

A prime number is a weak prime number for the base if it always loses its prime number property when any single digit is changed to any other digit (written in the base ). ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle b}$ ${\displaystyle p}$ ${\displaystyle b}$

Examples of weak prime numbers in other number systems

• The prime number is a weak base prime because: ${\displaystyle p=436_{7}={\underline {4}}\cdot 7^{2}+{\underline {3}}\cdot 7^{1}+{\underline {6}}\cdot 7^{0}=196+21+6=223}$ ${\displaystyle b=7}$

If you change one of the three digits in the base , you only get composite numbers, which are therefore not prime numbers: ${\displaystyle b=7}$

0 36 ₇ , 1 36 ₇ , 2 36 ₇ , 3 36 ₇ , 5 36 ₇ , 6 36 ₇

4 0 6 ₇ , 4 1 6 ₇ , 4 2 6 ₇ , 4 4 6 ₇ , 4 5 6 ₇ , 4 6 6 ₇

43 0 ₇ , 43 1 ₇ , 43 2 ₇ , 43 3 ₇ , 43 4 ₇ , 43 5 ₇

Overall, the above list gives composite numbers. ${\displaystyle 6\cdot 3=24}$

Representing all of the above 24 numbers, the number is checked for its primality: ${\displaystyle 433_{7}}$

${\displaystyle 433_{7}={\underline {4}}\cdot 7^{2}+{\underline {3}}\cdot 7^{1}+{\underline {3}}\cdot 7^{0}=196+21+3=220\not \in \mathbb {P} }$is not a prime number. Checking all other 23 numbers above works in the same way.

• The following table shows the smallest weak prime numbers for the base (sequence A186995 in OEIS ): ${\displaystyle b\leq 16}$

Base ${\displaystyle b}$	weak primes to the base , written to the base ${\displaystyle b}$ ${\displaystyle b}$	Conversion into the decimal system
1	${\displaystyle 11_{1}}$	${\displaystyle {\underline {1}}\cdot 1^{1}+{\underline {1}}\cdot 1^{0}=1+1=2}$
2	${\displaystyle 1111111_{2}}$	${\displaystyle {\underline {1}}\cdot 2^{6}+{\underline {1}}\cdot 2^{5}+{\underline {1}}\cdot 2^{4}+{\underline {1}}\cdot 2^{3}+{\underline {1}}\cdot 2^{2}+{\underline {1}}\cdot 2^{1}+{\underline {1}}\cdot 2^{0}=64+32+16+8+4+2+1=127}$
3	${\displaystyle 2_{3}}$	${\displaystyle {\underline {2}}\cdot 3^{0}=2}$
4th	${\displaystyle 11311_{4}}$	${\displaystyle {\underline {1}}\cdot 4^{4}+{\underline {1}}\cdot 4^{3}+{\underline {3}}\cdot 4^{2}+{\underline {1}}\cdot 4^{1}+{\underline {1}}\cdot 4^{0}=256+64+48+4+1=373}$
5	${\displaystyle 313_{5}}$	${\displaystyle {\underline {3}}\cdot 5^{2}+{\underline {1}}\cdot 5^{1}+{\underline {3}}\cdot 5^{0}=75+5+3=83}$
6th	${\displaystyle 334155_{6}}$	${\displaystyle {\underline {3}}\cdot 6^{5}+{\underline {3}}\cdot 6^{4}+{\underline {4}}\cdot 6^{3}+{\underline {1}}\cdot 6^{2}+{\underline {5}}\cdot 6^{1}+{\underline {5}}\cdot 6^{0}=23328+3888+864+36+30+5=28151}$
7th	${\displaystyle 436_{7}}$	${\displaystyle {\underline {4}}\cdot 7^{2}+{\underline {3}}\cdot 7^{1}+{\underline {6}}\cdot 7^{0}=<196+21+6=223}$
8th	${\displaystyle 14103_{8}}$	${\displaystyle {\underline {1}}\cdot 8^{4}+{\underline {4}}\cdot 8^{3}+{\underline {1}}\cdot 8^{2}+{\underline {0}}\cdot 8^{1}+{\underline {3}}\cdot 8^{0}=4096+2048+64+0+3=6211}$
9	${\displaystyle 3738_{9}}$	${\displaystyle {\underline {3}}\cdot 9^{3}+{\underline {7}}\cdot 9^{2}+{\underline {3}}\cdot 9^{1}+{\underline {8}}\cdot 9^{0}=2187+567+27+8=2789}$
10	${\displaystyle 294001_{10}}$	${\displaystyle {\underline {2}}\cdot 10^{5}+{\underline {9}}\cdot 10^{4}+{\underline {4}}\cdot 10^{3}+{\underline {0}}\cdot 10^{2}+{\underline {0}}\cdot 10^{1}+{\underline {1}}\cdot 10^{0}=200000+90000+4000+0+0+1=294001}$
11	${\displaystyle 2573_{11}}$	${\displaystyle {\underline {2}}\cdot 11^{3}+{\underline {5}}\cdot 11^{2}+{\underline {7}}\cdot 11^{1}+{\underline {3}}\cdot 11^{0}=2662+605+77+3=3347}$
12	${\displaystyle 6B8AB77_{12}}$	${\displaystyle {\underline {6}}\cdot 12^{6}+{\underline {11}}\cdot 12^{5}+{\underline {8}}\cdot 12^{4}+{\underline {10}}\cdot 12^{3}+{\underline {11}}\cdot 12^{2}+{\underline {7}}\cdot 12^{1}+{\underline {7}}\cdot 12^{0}=17915904+2737152+165888+17280+1584+84+7=20837899}$
13	${\displaystyle 2216_{13}}$	${\displaystyle {\underline {2}}\cdot 13^{3}+{\underline {2}}\cdot 13^{2}+{\underline {1}}\cdot 13^{1}+{\underline {6}}\cdot 13^{0}=4394+338+13+6=4751}$
14th	${\displaystyle C371CD_{14}}$	${\displaystyle {\underline {12}}\cdot 14^{5}+{\underline {3}}\cdot 14^{4}+{\underline {7}}\cdot 14^{3}+{\underline {1}}\cdot 14^{2}+{\underline {12}}\cdot 14^{1}+{\underline {13}}\cdot 14^{0}=6453888+115248+19208+196+168+13=6588721}$
15th	${\displaystyle 9880E_{15}}$	${\displaystyle {\underline {9}}\cdot 15^{4}+{\underline {8}}\cdot 15^{3}+{\underline {8}}\cdot 15^{2}+{\underline {0}}\cdot 15^{1}+{\underline {14}}\cdot 15^{0}=455625+27000+1800+0+14=484439}$
16	${\displaystyle D2A45_{16}}$	${\displaystyle {\underline {13}}\cdot 16^{4}+{\underline {2}}\cdot 16^{3}+{\underline {10}}\cdot 16^{2}+{\underline {4}}\cdot 16^{1}+{\underline {5}}\cdot 16^{0}=851968+8192+2560+64+5=862789}$

Properties of weak prime numbers in other number systems

• In order to determine whether a -digit prime number is a weak prime number as a base , one must control numbers, whether they are compound or not. Only when all numbers are put together is the -digit prime number actually a weak base prime . ${\displaystyle k}$ ${\displaystyle b}$ ${\displaystyle (b-1)\cdot k}$ ${\displaystyle (b-1)\cdot k}$ ${\displaystyle k}$ ${\displaystyle b}$
• Be a base. Then there are infinitely many weak prime numbers on this basis . ${\displaystyle b\in \mathbb {N} }$ ${\displaystyle b}$

Proof: see by Terence Tao from 2011.

Similar constructs

A similar construct represent the trunkierbaren prime numbers (from the English truncatable prime ) is from these prime numbers, any number of places disconnect without being lost their Primeigenschaft.:

• Linkstrunkierbare primes ( Left-truncatable primes ) (sequence A024785 in OEIS ) z. B. 1367 - 367, 67 and 7 would also be prime.
• Rechtstrunkierbare primes ( Right-truncatable primes ) (sequence A024770 in OEIS ), for example. B. 3739 - 373, 37 and 3 would also be prime.
• Two-sided primes that can be truncated on both sides (sequence A020994 in OEIS ) - in the strict definition of bilateral digit separability, there are only 15 prime numbers with this property:

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397

Web links

• Eric W. Weisstein : Weakly Prime . In: MathWorld (English).
• Weakly Primes in the Department of Mathematics of the Missouri State University (English)

Individual evidence

1. ^ NJA Sloane : Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes. OEIS , accessed December 10, 2018 .  
2. ↑ Weakly Primes. primepuzzles.net, 2012, accessed December 10, 2018 (largest known weak prime). 
3. ↑ ^{a b} Terence Tao : A remark on primality testing and decimal expansions . In: Journal of the Australian Mathematical Society . 91, No. 3, February 22, 2008. arxiv : 0802.3361 . doi : 10.1017 / S1446788712000043 .
4. ↑ Eric W. Weisstein : Truncatable Prime . In: MathWorld (English).