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Difference between revisions of "Home prime"

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The '''home prime''' of ''n'', denoted by HP(''n'') is found by concatenating the [[Prime number|prime]] [[factor]]s of ''n'', repeatedly, until a prime is reached. The notion of a home prime depends on the [[base]] (except in the case where ''n'' itself is prime).
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The '''home prime''' of ''n'', denoted by HP(''n'') is found by concatenating the [[prime]] [[factor]]s of ''n'', repeatedly, until a prime is reached. The notion of a home prime depends on the [[base]] (except in the case where ''n'' itself is prime).
  
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Notations for other bases of 10 are HP<sub>base</sub>(n) for base ''b''.
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While it is expected that every ''n'' in every base has a home prime, experimental evidence indicates that these chains can get quite long.
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==Base 10==
 
For example, the '''home prime''' of 25 is 773 because
 
For example, the '''home prime''' of 25 is 773 because
:25 = 5&times;5
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:25 = 5 &times; 5
:55 = 5&times;11
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:55 = 5 &times; 11
:511 = 7&times;73
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:511 = 7 &times; 73
 
and finally 773 is prime.
 
and finally 773 is prime.
  
 
The length of such chains is also of interest, in this case #HP(25) = 3.
 
The length of such chains is also of interest, in this case #HP(25) = 3.
  
While it is expected that every ''n'' in every base has a home prime, experimental evidence indicates that these chains can get quite long.
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==Base 2==
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HP<sub>2</sub>(10):
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:10<sub>10</sub> = 2 &times; 5 (10 &times; 101)
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:21<sub>10</sub> = 3 &times; 7 (11 &times; 111)
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:31 prime
  
 
==See also==
 
==See also==
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==External links==
 
==External links==
*[https://en.wikipedia.org/wiki/Home_prime Home prime at Wikipedia]
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*[[Wikipedia:Home prime|Home prime]]
[[Category:Math]]
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[[Category:Number]]

Latest revision as of 13:22, 6 March 2019

The home prime of n, denoted by HP(n) is found by concatenating the prime factors of n, repeatedly, until a prime is reached. The notion of a home prime depends on the base (except in the case where n itself is prime).

Notations for other bases of 10 are HPbase(n) for base b.

While it is expected that every n in every base has a home prime, experimental evidence indicates that these chains can get quite long.

Base 10

For example, the home prime of 25 is 773 because

25 = 5 × 5
55 = 5 × 11
511 = 7 × 73

and finally 773 is prime.

The length of such chains is also of interest, in this case #HP(25) = 3.

Base 2

HP2(10):

1010 = 2 × 5 (10 × 101)
2110 = 3 × 7 (11 × 111)
31 prime

See also

External links