# Talk:Deceptive numbers

Since the repunit ends in 1 5 can not be a factor of n--Nigel Galloway (talk) 11:44, 12 February 2022 (UTC)

Since the repunit is odd, 2 cannot be a factor of **n**. And because the digit sum of the repunit is **n - 1**, 3 cannot be a factor of **n** (an integer is divisible by 3, if its digit sum is divisible by 3, but **n - 1** and **n** cannot both be divisible by 3).

All non-primes below 7^{2} are divisible by 2, 3, or 5. So it is sufficient to start the search at 49. --Querfeld (talk) 14:01, 24 September 2023 (UTC)

Regarding the repunit **R _{n-1}**, the equation

**R**is fulfilled. And if it is ensured, that

_{n-1}* 9 = 10^{n-1}- 1**n**is not divisible by 3, then if

**R**is divisible by

_{n-1}* 9**n**, also

**R**is divisible by

_{n-1}**n**. In that case, the check of

**R**can be simplified to

_{n-1}mod n = 0**10**, which can be calculated via modular exponentiation, so that big integers are not required for this task. --Querfeld (talk) 19:09, 26 September 2023 (UTC)

^{n-1}mod n = 1- Yep, that certainly works! (actually nine entries were already doing just that last year) --Petelomax (talk) 23:52, 26 September 2023 (UTC)
- Indeed (actually, the OCaml, Python, and C implementations have been written by me; translations followed), I just wanted to explain, how this works — to encourage implementations in languages, that don't provide native BigInt support (32-bit precision is sufficient for the first 50
*deceptive numbers*). And the principle has already been shown on the OEIS page (just that they*mod*by**n * 9**, instead of ruling out factor 3 in advance). But it is often faster to do the Fermat primality test*before*the full primality check. --Querfeld (talk) 10:59, 27 September 2023 (UTC)

- Indeed (actually, the OCaml, Python, and C implementations have been written by me; translations followed), I just wanted to explain, how this works — to encourage implementations in languages, that don't provide native BigInt support (32-bit precision is sufficient for the first 50

With the wp:Fermat primality test being **a ^{n-1} mod n = 1**, where

**gcd(a, n) = 1**, it becomes obvious, that the

*deceptive numbers*are a subset of the Fermat pseudoprimes to base

**a = 10**, with the multiples of 3 filtered out (

**10**already rules out the multiples of 2 and 5). --Querfeld (talk) 19:09, 26 September 2023 (UTC)

^{x}