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July 23

Perfect numbers one less than primes

The only perfect number that is one more than a prime number is 6, because subtracting one from any odd perfect number (should such numbers exist) obviously gives an even composite number, and any even perfect number other than 6 is congruent to 1 mod 3.

In contrast, are there many (obviously even) perfect numbers that are one less than primes? The numbers 7(=6+1) and 29(=28+1) are prime, but 497(=496+1) is composite because it is divisible by 7 (which happens to be the number of days in a week, and this also shows that the Gregorian calendar repeats with a cycle of 400 years: July 4, 2176 will be a Thursday just as it was in 1776). GeoffreyT2000 (talk) 01:20, 23 July 2019 (UTC)[reply]

Apparently not, see OEIS: A061644. Note that you can easily eliminate about half the candidates because if p≡2 (mod 3) then 2^p-1(2^p-1)+1 is divisible by 7. The OEIS entry has a more sophisticated criterion which I haven't investigated yet. --RDBury (talk) 04:27, 23 July 2019 (UTC)[reply]

I'm not a mathematician (I didn't know what a perfect number was till I looked it up) but the reasoning here leaves me baffled. Why does the fact that 497 is obviously divisible by seven (it's (10 x 7² + 7) show that the Gregorian calendar repeats with a cycle of 400 years? 2A00:23C5:C708:8C00:B0C8:D69:FA32:D1C8 (talk) 10:01, 24 July 2019 (UTC)[reply]

400 Gregorian years are 400*365 non-leap days and 97 leap days (29 February). 400*365+97=400*364+497, where 364=7*52. —Kusma (t·c) 10:26, 24 July 2019 (UTC)[reply]

For greater clarity: the Gregorian calendar specifies a 400-year cycle of leap years. The multiples of 7 come in when we consider days of the week. So for example July 24, 2019, is a Wednesday; then if the Gregorian calendar is still in use, July 24, 2419, will also be a Wednesday. That is what was meant by the calendar repeating. --69.159.11.113 (talk) 20:59, 24 July 2019 (UTC)[reply]

... although the fact that 497 also happens to be one greater than a perfect number is a numerical coincidence. Gandalf61 (talk) 09:10, 25 July 2019 (UTC)[reply]