Vipul: Created page with "==Definition== ===For arbitrary base=== A prime $p$ is termed a '''full reptend prime''' to base $b$ (where $b$ is a positive integer greate..."

2012-01-03T20:27:51Z

Created page with "==Definition== ===For arbitrary base=== A prime <math>p</math> is termed a '''full reptend prime''' to base <math>b</math> (where <math>b</math> is a positive integer greate..."

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==Definition==

===For arbitrary base===

A prime <math>p</math> is termed a '''full reptend prime''' to base <math>b</math> (where <math>b</math> is a positive integer greater than 1) if the following equivalent conditions are satisfied:

# <math>b</math> is a [[defining ingredient::primitive root]] modulo <math>p</math>.
# The base <math>b</math> expansion of <math>1/p</math> has repeating block of length <math>p - 1</math>.
# The number <math>(b^{p-1} - 1)/p</math> is a [[defining ingredient::cyclic number]] in base <math>b</math>.

===Default of base 10===

By default, the term '''full reptend prime''' is used for a prime that is a full reptend prime in base 10.

==Facts==

For a fixed base <math>b</math>, there exists a finite number dependent on <math>b</math> such that whether or not <math>p</math> is a full reptend prime mod <math>b</math> depends only on the congruence class of <math>p</math> modulo that finite number. When <math>b</math> itself is an odd prime, this number is <math>4b</math>.

Full reptend prime - Revision history

Vipul: Created page with "==Definition== ===For arbitrary base=== A prime p is termed a '''full reptend prime''' to base b (where b is a positive integer greate..."

Vipul: Created page with "==Definition== ===For arbitrary base=== A prime $p$ is termed a '''full reptend prime''' to base $b$ (where $b$ is a positive integer greate..."