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Prove using induction that every positive integer $>23$ can be expressed in the form $=5 +7$ for appropriately chosen non-negative integers $a, b$.

What should I use? Mathematical Induction or Strong Induction?

Can anyone help me solving this question? It confuses me a lot. Any help will be appreciated.

Jean Marie
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dash_warrior
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  • If you are in doubt between the two, use strong induction. Anyways, worst that can happen with using $[P(n)\to P(n+1)]$ induction instead of $[(\forall m<n,,P(m))\to P(n)]$ induction is that you might need to switch to proving $[Q(n)\to Q(n+1)]$, where $Q(n)$ stands for "$\forall m<n, P(m)$". Worst that can happen with both is that you might need a different idea altogeher. –  Sep 18 '20 at 19:09
  • @dash_warrior Welcome to Math SE. FYI, please see coin problem. – John Omielan Sep 18 '20 at 19:16
  • For an insight into this type of problem, it is interesting to know that it is the "Frobenius coin problem"; I have found interesting to modify your very neutral title "I have aproblem with mathematical induction" with this name ; besides, take a look at https://math.stackexchange.com/q/8241 among many references. – Jean Marie Sep 18 '20 at 20:32

1 Answers1

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Checking for $n=24$-$28$:

$24=5\times2+7\times2, 25=5\times5, 26=5+7\times3, 27=5\times4+7, 28=7\times4$.

Assume that for $n=k$, $k$ can be expressed as $5a+7b$.

Considering $n=k+5$, $k+5=5a+7b+5=5(a+1)+7b$.

Therefore, since $n=24$-$28$ is true, so is $n=29$-$33$, and thus so is $n=34$-$38$, and thus, in this manner, so is for the entirety of integers greater than $23$.

StrongestStrike
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