Unique prime - Citizendia

In mathematics, a unique prime is a certain kind of prime number. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 A prime p ≠ 2, 5 is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1 / p, is equivalent to the period length of the reciprocal of q, 1 / q. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which Unique primes were first described by Samuel Yates in 1980. Samuel Yates is a Mathematician who first described Unique primes in the 1980s

It can be shown that a prime p is of unique period n if and only if there exists a natural number c such that

$\frac{\Phi_n(10)}{\gcd(\Phi_n(10),n)} = p^c$

where Φ_n(x) is the n-th cyclotomic polynomial. ↔ In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power At present, more than fifty unique primes or probable primes are known. In Number theory, a probable prime (PRP is an Integer that satisfies a specific condition also satisfied by all Prime numbers. However, there are only twenty-three unique primes below 10¹⁰⁰. The following table gives an overview of all 23 unique primes below 10¹⁰⁰ (sequence A040017 in OEIS) and their periods (sequence A051627 in OEIS):

Period length	Prime
1	3
2	11
3	37
4	101
10	9,091
12	9,901
9	333,667
14	909,091
24	99,990,001
36	999,999,000,001
48	9,999,999,900,000,001
38	909,090,909,090,909,091
19	1,111,111,111,111,111,111
23	11,111,111,111,111,111,111,111
39	900,900,900,900,990,990,990,991
62	909,090,909,090,909,090,909,090,909,091
120	100,009,999,999,899,989,999,000,000,010,001
150	10,000,099,999,999,989,999,899,999,000,000,000,100,001
106	9,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
93	900,900,900,900,900,900,900,900,900,900,990,990,990,990,990,990,990,990,990,991
134	909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,090,909,091
294	142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143
196	999,999,999,999,990,000,000,000,000,099,999,999,999,999,000,000,000,000,009,999,999,999,999,900,000,000,000,001

The prime with period length 294 is similar to the reciprocal of 7 (0. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences ---- In mathematics Three is the first odd Prime number, and the second smallest prime 37 ( thirty-seven) is the Natural number following 36 and preceding 38. 142857142857142857. . . )

Just after the table, the twenty-fourth unique prime has 128 digits and period length 320. It can be written as (9₃₂0₃₂)₂ + 1, where a subscript number n indicates n consecutive copies of the digit or group of digits before the subscript. Though they are rare, based on the occurrence of repunit primes and probable primes, it is conjectured strongly that there are infinitely many unique primes. In Recreational mathematics, a repunit is a Number like 11, 111, or 1111 that contains only the digit 1 In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of

As of 2006 the repunit R₈₆₄₅₃ is the largest known probable unique prime. Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar.

In 1996 the largest proven unique prime was (10¹¹³² + 1)/10001 or, using the notation above, (99990000)₁₄₁+ 1. Its period of reciprocal is 2264. The record has been improved many times since 2000. As of 2008 the largest proven unique prime has 7200 digits, proved by Raffi Chaglassian in 2005. ^[1]

References

^ The Top Twenty Unique; Chris Caldwell

External links

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