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

Take an even positive integer .  is either , , or . Notice that the numbers , , , ... , and in general  for nonnegative  are odd composites. We now have 3 cases:

If  and is ,  can be expressed as  for some nonnegative . Note that  and  are both odd composites.

If  and is ,  can be expressed as  for some nonnegative . Note that  and  are both odd composites.

If  and is ,  can be expressed as  for some nonnegative . Note that  and  are both odd composites.

Clearly, if , it can be expressed as a sum of 2 odd composites. However, if , it can also be expressed using case 1, and if , using case 3.  is the largest even integer that our cases do not cover. If we examine the possible ways of splitting  into two addends, we see that no pair of odd composites add to . Therefore,  is the largest possible number that is not expressible as the sum of two odd composite numbers.

From ProofWiki

Let $k \in \Z_{>0}$ be a (strictly) positive integer.

The largest even integer which cannot be expressed as the sum of $2 k$ odd positive composite integers is $18 k + 20$.

Proof

Let $n$ be an even integer greater than $18 k + 20$.

Then $n - 9 \paren {2 k - 2}$ is an even integer greater than $18 k + 20 - 9 \paren {2 k - 2} = 38$.

By Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers, every even integer greater than $38$ can be expressed as the sum of $2$ odd positive composite integers.

Thus $n - 9 \paren {2 k - 2}$ can be expressed as the sum of $2$ odd positive composite integers.

So, let $a$ and $b$ be odd composite integers such that $a + b = n - 9 \paren {2 k - 2}$.

Then:

$9 \paren {2 k - 2} + a + b = n$

This is an expression for $n$ as the sum of $2 k$ odd positive composite integers, in which $2 k - 2$ of them are occurrences of $9$.

Thus such an expression can always be found for $n > 18 k + 20$.

$\Box$


It remains to be shown that $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.

The $k = 1$ case is demonstrated in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.


Now suppose $k \ge 2$.

The two smallest odd positive composite integers are $9$ and $15$.

Suppose $18 k + 20$ is expressible as the sum of $2 k$ odd positive composite integers.

Then at least $2 k - 3$ of them are $9$'s, because:

$9 \paren {2 k - 4} + 4 \times 15 = 18 k + 24 > 18 k + 20$

Then the problem reduces to finding an expression of $18 k + 20 - 9 \paren {2 k - 3} = 47$ as the sum of $3$ odd positive composite integers.

The first few odd positive composite integers are:

$9, 15, 21, 25, 27, 33, 35, 39, 45$

Their differences with $47$ are:

$38, 32, 26, 22, 20, 14, 12, 8, 2$

The integers above are in the set of integers not expressible as a sum of $2$ odd positive composite integers.

The full set of these can be found in Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers.

Thus $18 k + 20$ is not expressible as the sum of $2 k$ odd positive composite integers.

This proves the result.

$\blacksquare$

Historical Note

According to the footnote to the presentation of the solution to Positive Even Integers not Expressible as Sum of 2 Composite Odd Numbers in 1990: Solution to Problem 1328 (Math. Mag. Vol. 63, no. 4: pp. 273 – 280)  www.jstor.org/stable/2690953, this particular result was deduced by the Shippensburg University Mathematical Problem Solving Group.

Sources

For $n \geq 39$, at least one of the numbers $n - 9, n - 15, n - 21, n-27, n- 33$ (which are all greater than $5$) must be divisible by $5$, hence composite. So you can limit your search to numbers $n \leq 38$.

As noted in other answers, it turns out that $38$ works. This can be checked as follows. If we had $38 = a + b$ where $a$ and $b$ were odd composite numbers, at least one of them, say $a$, would have to be $\leq 19$. So it's enough to check that $38 - 9 = 29$ and $38 - 15 = 23$ are prime.

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

Solution 1

Take an even positive integer . is either

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, or
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. Notice that the numbers ,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, ... , and in general
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
for nonnegative are odd composites. We now have 3 cases:

If

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and is
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, can be expressed as
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
for some nonnegative . Note that and
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
are both odd composites.

If

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and is
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, can be expressed as
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
for some nonnegative . Note that
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
are both odd composites.

If

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and is
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, can be expressed as
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
for some nonnegative . Note that
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
are both odd composites.


Clearly, if

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, it can be expressed as a sum of 2 odd composites. However, if
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, it can also be expressed using case 1, and if
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, using case 3.
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
is the largest even integer that our cases do not cover. If we examine the possible ways of splitting
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
into two addends, we see that no pair of odd composites add to
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. Therefore,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
is the largest possible number that is not expressible as the sum of two odd composite numbers.

Solution 2

Let be an integer that cannot be written as the sum of two odd composite numbers. If

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, then
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
must all be prime (or
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, which yields
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
which does not work). Thus
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
form a prime quintuplet. However, only one prime quintuplet exists as exactly one of those 5 numbers must be divisible by 5.This prime quintuplet is
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
and
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, yielding a maximal answer of 38. Since
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
, which is prime, the answer is
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
.

Solution 3 (bash)

Let

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
be an even integer. Using the Chicken McNugget Theorem on the two smallest odd composite integers that are relatively prime from each other, 9 and 25, show that the maximum is 191, and the maximum even integer is 190 or lower. We use the fact that sufficiently high multiples of 6, 10, 14, 22, etc. can be represented as
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. We bash each case until we find one that works.

Solution 4 (easiest)

The easiest method is to notice that any odd number that ends is a 5 is a composite (except for 5 itself). This means that we will have 15, 25, 35, etc... no matter what. What it also means is that if we look at the end digit, if 15 plus another number will equal that number, then any number that has that same end digit can be added by that same number plus a version of 15, 25, 35...

For example, let's say we assume our end digit of the number is 4. If we have 5 as one of our end digits, then 9 must be the end digit of the other number. If we go down our list of numbers that end with a 9 and is composite, we will stumble upon the number 9 itself. That means that the number 15+9 is able to be written in a composite form, but also anything that ends with a 4 and is above 15+9. Hence the largest number that ends with a 4 that satisfies the conditions is 14.

If you list out all the numbers, you will notice that 33 is the largest number where the last digit is not repeated (13 and 23 are not composite). That means that 33+15 and anything else that ends with a 3 is bad, so the largest number that satisfies the conditions is the largest number that ends with a 8 and is below 48. That number would be

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

Solution 5

Claim: The answer is

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

Proof: It is fairly easy to show 38 can't be split into 2 odd composites.

Assume there exists an even integer

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
that m can't be split into 2 odd composites.

Then, we can consider m modulo 5.

If m = 0 mod 5, we can express m = 15 + 5k for some integer k. Since

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
so
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. Thus, 5k is composite. Since 15 is odd and composite, m is even, 5k should be odd as well.

If m = 1 mod 5, we can express m = 21 + 5k for some integer k. Since

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
so
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. Thus, 5k is composite. Since 21 is odd and composite, m is even, 5k should be odd as well.

If m = 2 mod 5, we can express m = 27 + 5k for some integer k. Since

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
so
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. Thus, 5k is composite. Since 27 is odd and composite, m is even, 5k should be odd as well.

If m = 3 mod 5, we can express m = 33 + 5k for some integer k. Since

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
so
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. Thus, 5k is composite. Since 33 is odd and composite, m is even, 5k should be odd as well.

If m = 4 mod 5, we can express m = 9 + 5k for some integer k. Since

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
,
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
so
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
. Thus, 5k is composite. Since 9 is odd and composite, m is even, 5k should be odd as well.

Thus, in all cases we can split m into 2 odd composites, and we get at a contradiction.

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

-Alexlikemath

Video Solution using Bashing

https://www.youtube.com/watch?v=n98zEG1-Hrs ~North America Math Contest Go Go Go

See also