07-19-2026, 11:11 PM
Full reptend prime
Summary
A full reptend prime (also called a long prime or full repetend prime) is a special type of prime number whose reciprocal produces a repeating decimal with the maximum possible length. For a prime number (p) in base 10, the decimal expansion of ($1/p$) repeats after exactly $(p-1)$ digits, creating a cyclic sequence of numbers. For example, ($1/7 = 0.142857142857...$), where the six-digit repeating block ($142857$) contains every possible cyclic rotation when multiplied by numbers from 1 to 6.
Mathematically, a prime is full reptend when the base (10 in decimal notation) is a primitive root modulo (p), meaning that powers of 10 generate all nonzero residues modulo (p). These primes have connections to modular arithmetic, cyclic numbers, and number theory, and examples include ($7, 17, 19, 23,$) and ($29$). Although many full reptend primes have been discovered, mathematicians still do not know a simple way to generate all of them. Artin’s conjecture on primitive roots suggests that approximately 37% of all primes should be full reptend primes in base 10, but this remains unproven.
ARTICLE
Summary
A full reptend prime (also called a long prime or full repetend prime) is a special type of prime number whose reciprocal produces a repeating decimal with the maximum possible length. For a prime number (p) in base 10, the decimal expansion of ($1/p$) repeats after exactly $(p-1)$ digits, creating a cyclic sequence of numbers. For example, ($1/7 = 0.142857142857...$), where the six-digit repeating block ($142857$) contains every possible cyclic rotation when multiplied by numbers from 1 to 6.
Mathematically, a prime is full reptend when the base (10 in decimal notation) is a primitive root modulo (p), meaning that powers of 10 generate all nonzero residues modulo (p). These primes have connections to modular arithmetic, cyclic numbers, and number theory, and examples include ($7, 17, 19, 23,$) and ($29$). Although many full reptend primes have been discovered, mathematicians still do not know a simple way to generate all of them. Artin’s conjecture on primitive roots suggests that approximately 37% of all primes should be full reptend primes in base 10, but this remains unproven.
ARTICLE
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│ KONSTANTINOS MICHAILIDIS │
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