Let's denote with

the repunit 111...111 made of n ones. It is known that every prime p>3 divides the repunit

. For example, 15 divides

. R.Francis & T.Ray call a composite number c deceptive if it has the same property, i.e., if it divides the repunit

. For example, 55 is deceptive because it divides

Francis & Ray have proved that there are infinite deceptive numbers since, if n is deceptive, then

is deceptive as well.

Definitions and Theorems

Definition 1: A repunit, Rn, is an integer consisting entirely of n “ones” in its dozenal representation. It is defined algebraically as (10n−1)/E

Corollary 2: If p ≠ 2,3,E is prime, then p|Rp−1

Definition 2: A composite number n satisfying the condition n|Rn−1 is called a deceptive prime.

The deceptive primes less than 10000 are 55, 77, E1, 101, 187, 275, 4X7, 777, 781, E55, 1001, 1111, 1117, 1221, 1515, 1691, 1771, 1887, 1E47, 3131, 3267, 3421, 3795, 39X1, 51X7, 5301, 6161, 7217, 7401, 7575, 7677, 7871, 7881, 8181, 8327, 8805, 8921, 9201, 9647, XX01, E1E1, E365, EX37

Theorem 3: If n is a deceptive prime, then Rn is also a deceptive prime.

Theorem 4: If p ≠ E is prime and Rp is composite, then Rp is a deceptive prime.

Definition 3: A prime p is called a primitive divisor of Rn if p|Rn but p ̸ |Rm for all m < n. For example, 11 is a primitive divisor of R2, also, 11|R10 but, since 11|R2, 11 is not a primitive divisor of R10. Note that EE01 is a primitive divisor of R10

Theorem 5: Let m > 1 be an integer. If c ≠ E is a primitive divisor of Rm and d is a primitive divisor of R2m, then cd is a deceptive prime.

Theorem 6: Let p ≠ E and q ≠ E be any two distinct primitive divisors of Rn, where n is odd. Then their product, pq, is a deceptive prime.

Conjectures

1. For every integer n > 1, the intersection of the set of deceptive primes and the set of strong pseudoprimes base n is infinite (if this conjecture is true, then the intersection of the set of deceptive primes and the set of (Fermat, Euler, Euler-Jacobi) pseudoprimes base n is also infinite, since all strong pseudoprimes base n are also Euler-Jacobi pseudoprimes base n, all Euler-Jacobi pseudoprimes base n are also Euler pseudoprimes base n, all Euler pseudoprimes base n are also Fermat pseudoprimes base n).

2. Almost all deceptive primes have a terminal digit of 1

3. For every digit d in {1, 5, 7, E}, infinitely many deceptive primes have a terminal digit of d

4. Every prime number except 2, 3, E, divides infinitely many deceptive primes.

5. Any deceptive prime is a divisor of a still larger deceptive prime (if this conjecture is true, then it divides infinitely many).

6. For all n > 1, there are infinitely many deceptive primes having exactly n distinct prime divisors.

7. All deceptive primes are deficient.

8. All deceptive primes are cubefree. (note that not all deceptive primes are squarefree, 1685^2 = 2518XX1 is deceptive prime, the conjecture that all deceptive primes are squarefree might be true in bases 19, 25, 3E, 42, 51, 60, 61, X6)

9. Numbers of the form 22n+1, n!±1, or 2n−1 cannot be elements of the set of deceptive primes. (note that this is not true for 2n+1, 26+1 = 55 is deceptive prime, and indeed the smallest deceptive prime)

X. For all n > 1, a deceptive prime exists having exactly n digits.

E. Infinitely many deceptive primes n exist in which n divides no repunit smaller than Rn−1

10. Though the set of twin primes is unclassified as to cardinality, there are no twin deceptive primes.

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