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February 13

Even perfect numbers in duodecimal

Is it true that every even perfect number other than 6 and 24 in duodecimal ends with the digits "54"? Remember that while the duodecimal number 24 may look like the number twenty-four, it really means twenty-eight, which, of course, we know is a perfect number.

As an example, consider the perfect number 496 (in decimal). In duodecimal, it becomes 354 (not to be confused with the decimal number three hundred and fifty-four). Notice that the last two digits here are "54". GeoffreyT2000 (talk) 06:01, 13 February 2020 (UTC)[reply]

Yes, it is true. Here is a simpler statement, not involving primality, that implies the one above about perfect numbers:

For all odd p, p ≥ 5, 2^2p−1 − 2^p−1 ≡ 2⁶ (mod 144).

This statement can easily be checked by checking it separately (using modular arithmetic) for each of the three congruence classes of odd p modulo 6. Note that 2⁶·2⁶ ≡ 2⁶ (mod 144). --Lambiam 08:39, 13 February 2020 (UTC)[reply]

The necessary arithmetic calculations can be somewhat simplified:

2^2p−1 − 2^p−1 ≡ 2⁶ (mod 144)

⇔ 2^2p+5 − 2^p+5 ≡ 2⁶ (mod 144)

⇔ 2⁴·(2^2p+1 − 2^p+1) ≡ 2⁴·4 (mod 2⁴·9)

⇔ 2^2p+1 − 2^p+1 ≡ 4 (mod 9)

⇔ 4·(2^2p−1 − 2^p−1) ≡ 4 (mod 9)

⇐ 2^2p−1 − 2^p−1 ≡ 1 (mod 9).

So the original statement follows by modular arithmetic from the fact that sufficiently large (≥ 28) even perfects end on a digit 1 when expressed nonally. I see no obvious way of avoiding case analysis. --Lambiam 13:58, 13 February 2020 (UTC)[reply]

Without case analysis. Let v = 2^p−1. Since p is odd, we can write it in the form 2q + 1, so then v = 4^q ≡ 1 (mod 3). Hence, v can be written in the form 3u + 1. Now, by elementary algebra,

2^2p−1 − 2^p−1

= (2^p−1)(2·2^p−1 − 1)

= v(2v − 1)

= (3u + 1)(6u + 1)

= 18u² + 9u + 1

≡ 1 (mod 9).

--Lambiam 19:20, 13 February 2020 (UTC)[reply]