# Largest Even Integer: Impossible Sum of Two Odd Composites

• MHB
• anemone
In summary, the largest even integer that cannot be expressed as the sum of two odd composite numbers is 4. This is because 4 is the smallest even integer that cannot be expressed as the sum of two odd composite numbers, and any larger even integer can be expressed as the sum of two odd composite numbers. However, any odd composite number can be expressed as the sum of two odd composite numbers, and there are no exceptions to the rule that every even integer can be expressed as the sum of two odd composite numbers. The significance of this number lies in its role as a boundary point for understanding the relationships between even integers and odd composite numbers and the patterns and properties of numbers and their sums.
anemone
Gold Member
MHB
POTW Director
Find the largest even integer which cannot be written as the sum of two odd composite numbers.

anemone said:
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

anemone said:
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.

## 1. What is the largest even integer that cannot be written as the sum of two odd composite numbers?

The largest even integer that cannot be written as the sum of two odd composite numbers is 4.

## 2. What are odd composite numbers?

Odd composite numbers are positive integers that are not prime and are not divisible by 2.

## 3. Why is it impossible to write the largest even integer as the sum of two odd composite numbers?

This is because the largest even integer is always divisible by 2, and the sum of two odd composite numbers will always be an odd number. Therefore, it is impossible for an even number to be the sum of two odd numbers.

## 4. Are there any exceptions to this rule?

No, there are no exceptions to this rule. The largest even integer cannot be written as the sum of two odd composite numbers.

## 5. How is this concept relevant in mathematics?

This concept is relevant in number theory and algebra, as it helps in understanding the properties of even and odd numbers and their relationships. It also helps in identifying prime and composite numbers and their sums.

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