Weakly prime number
Encyclopedia
In number theory
, a prime number
is called weakly prime if it becomes composite
when any one of its digits is changed to every single other digit. Decimal
digits are usually assumed.
The first weakly prime numbers are:
For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-b
number with n digits must produce (b−1) × n composite numbers when a digit is changed.
In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17−17)/99 + 21686652. This is the largest known weakly prime number .
There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes.
The smallest weakly prime base-b number for b = 2 to 10 is:
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
is called weakly prime if it becomes composite
Composite number
A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
when any one of its digits is changed to every single other digit. Decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
digits are usually assumed.
The first weakly prime numbers are:
- 294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, ...
For the first of these, each of the 54 numbers 094001, 194001, 394001, ..., 294009 are composite. A weakly prime base-b
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
number with n digits must produce (b−1) × n composite numbers when a digit is changed.
In 2007 Jens Kruse Andersen found the 1000-digit weakly prime (17−17)/99 + 21686652. This is the largest known weakly prime number .
There are infinitely many weakly prime numbers in any base. Furthermore, for any fixed base there is a positive proportion of such primes.
The smallest weakly prime base-b number for b = 2 to 10 is:
- 11111112 = 127
- 23 = 2
- 113114 = 373
- 3135 = 83
- 3341556 = 28151
- 4367 = 223
- 141038 = 6211
- 37389 = 2789
- 29400110 = 294001