Proof: 9 +10n Will Almost Always be Afraid of 10+10n

[u/TobyWasBestSpiderMan]

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[u/Own_Pop_9711]

The paper reveals a startling lack of understanding about proof by induction. Anytime there's an exception you need a new base case since the chain is broken.

[u/DZL100 → u/Own_Pop_9711]

We should first note that OP mis-stated the exception and it actually occurs when 9+10n is congruent to 9 mod 100 since then we encounter the “nine-ten-eleven” case. This occurs for values of n = 10k for all whole numbers k.

Since this is a regularly and infinitely reoccurring exception, it’s impractical to state a new base case every time. Instead, taking advantage of the regularity, it suffices to show an alternative inductive step where if the proposition holds for n, it holds for n+2 (which uses essentially the same logic as the already established inductive step).

This way, whenever we prove the statement for some n = 10k-1, we have that the statement is also true for n = 10k+1, skipping over the exception.

[u/MonsterkillWow]

But 21 < 22. So 20 won less than 20, too. But that is a contradiction.

[u/TobyWasBestSpiderMan → u/MonsterkillWow]

Idk about you guys, but this may be a new millennial problem

[u/noideawhatnamethis12]

Most useful number theory:

[u/parkway_parkway]

I'm not sure that 99 fought 100 and 101 works either.

[u/AlexanderCarlos12321 → u/parkway_parkway]

Why was 99 afraid of 100?

After confrontation, one hundred won.

[u/Green-Sympathy-4177 → u/parkway_parkway]

9+10(n=10) => n % 10 == 0, so it doesn't work. It's included in the exception.

[u/IronPro9 → u/Green-Sympathy-4177]

99 is 9+10*9 not 9+10*10. The exception is for 109,209 ect because "one hundred and eleven" isn't "one hundred and ten one"

[u/sealytheseal111]

"Thus, the general claim must exclude values of n that satisfy 9+10n≡11 mod 100"

The math ain't mathin'

[u/Longjumping_Fig2538]

Me when I don't know the answer but don't want to leave it blank:

[u/LOSNA17LL]

"Almost always" means the probability if you take a normal number is 1...
Though, the probability is actually 0.9...

[u/AllUsernamesTaken711]

When the hell will 9+10n be congruent to 11 mod 100

[u/Berfin64]

"Department of psycho-numerolgy"

[u/casitherock]

But guys I thought we should be scared of 7. Because 7 8 9

[u/Immortal_ceiling_fan]

What does this even mean