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[SOLVED] Largest even integer

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,642
Find the largest even integer which cannot be written as the sum of two odd composite numbers.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,121
Find the largest even integer which cannot be written as the sum of two odd composite numbers.
I must be missing something. Say we have a, b, c, d are all odd primes. Then e = ab + cd. But there is no largest prime so how can e be bounded?

-Dan
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,697
Find the largest even integer which cannot be written as the sum of two odd composite numbers.
I will use the notation $*5$ to denote any positive integer ending in $5$, apart from the number $5$ itself. So $*5$ could be $15,25,35,\ldots$. Notice that any number of the form $*5$ is odd and composite.

The smallest odd composite numbers are $9,15,21,25,27,33,\ldots$.

If an even integer ends in $0$ and is greater than $20$ then it is of the form $15 + *5$.

If an even integer ends in $2$ and is greater than $32$ then it is of the form $27 + *5$.

If an even integer ends in $4$ and is greater than $14$ then it is of the form $9 + *5$.

If an even integer ends in $6$ and is greater than $26$ then it is of the form $21 + *5$.

If an even integer ends in $8$ and is greater than $38$ then it is of the form $33 + *5$.

The largest even number not included in any of those categories is $38$. You can easily verify that $38$ cannot be expressed as the sum of two odd composite numbers. So it is the largest such even number.