Brainsteem #5 Prize Maths Puzzle [Win 2SBD+1SBD]
Let S(n) be the sum of the divisors of a positive integer n. This includes both 1 and n itself. For example, S(6) = 12.
Find all solutions to the equation:
S(n + 2) = S(n) + 2
where 0 < n < 100
Remember that these questions should be done by hand and without a calculator!
The art of solving such questions is to quickly find some pattern or underlying structure so that you do not have to calculate S(n) for all values of n.
As always, the top prize requires a full solution and a brief method of the steps taken.
Any questions, just ask!
image: designed myself
The Prizes
Earnings from each Brainsteem Puzzle will go towards the next Prizes.
For this Brainsteem # 5 the Prizes are:
2 SBD for the first correct answer that includes a valid method
1 SBD for one other correct answer that will be chosen randomly. That means you can still win this prize even if the answer may already be visible!
Please upvote! Resteem if you think this is a great puzzle! I will upvote all correct answers and any other interesting comments. To be fair to everyone, the prize closes after 7 days. Winners will be notified after that date and usernames posted with a solution.
Enjoy!
nice one, let me grab my note pad, smells a lot like induction ;)
Edit: I read the question wrong, I thought this is a proof. I thought i had found a flaw at S(6)=/=S(4)+S(2). xD
this one is tough, nice puzzle ^_^
If you don't come out right now S(n), I will kill S(2) right now! You have 1 minute!
Solved. Check after 1 minute.
The Martian
Sorry, discovered an important error! The equation is S(n+2)=S(n)+2.
Have edited the question.
oh right... I have been struggling at S(n+2) = s(n) + 3 yesterday... let me try again now~
Yeah, luckily I read it again. I think my brain thought it was a nice distributive property while typing!!
As I was re-solving it in my head I thought: huh! S(2)=3, this doesn't work!
yup I understand that, S(n+2) = S(n)+S(2) is definitely more beautiful XD
anyway, I agree with guyverckw 's answer (3,5,11,17,41,47), but I am thinking of a rigorous method to prove the equality only holds when n and n+2 are both prime... can't think of it yet
it is easy to prove if n and n+2 are both prime, S(n+2) = S(n)+2. but the converse may not be true...
Answers: 3, 5, 11, 17, 41, 47
Find the answers by basically list out first 13 equations and spot the patterns, which may not be the only patterns of this series.
If there is a more mathematical way of getting the answers, will be more than happy to know. Thanks.
The Winner are:
First Prize to @guyverckw - you may claim either the 2 SBD or a free randowhale vote, please let me know within 2 days or I will process the SBD payment.
Second Prize to @kenchung of 1 SBD - on its way!
Check out the current live Brainsteems!
thanks man! but I would like to know the model answer from you, is there any better way of doing this?
Brainsteem #6 posted now:
https://steemit.com/education/@rycharde/brainsteem-6-prize-maths-puzzle-win-2sbd-1sbd
Enjoy!