that can not be written as the sum of two odd composite numbers (with proof), or prove that there are infinitely many such even integers with this property.
Hint: There is a largest and it’s not too large.
Consider:
40 = 15 + 25
42 = 15 + 27
44 = 35 + 9
46 = 25 + 21
48 = 15 + 33
Each of these has as one of the terms of the addition a multiple of 5.
We can add any multiple of 10 to said multiple of 5 to get another
pair of composite odd numbers which has an even sum,
and which is 10 * n larger than these numbers.
So all numbers 40 and above can be written as the sum of two odd composites
(often in many different ways, but we only need this one).
Now consider 38:
The odd composite numbers < 38 are
9,15,21,25,27,33
and no two of them add up to 38.
So the answer is 38.
We will prove this property by contratiction!
Assume the contrary means there is only finite no of such numbers..
Let largest be n+m where n,m are odd composite no/
And so n+m=even no and also largest among all having such property..
But we could find 3n and 3m which also odd composite intergers
So 3n+3m=3[n+m] belongs to the integer set
We could find some 5n,5m,7n,7m,9n,9m……………so on
But contradicts to our fact that n+m is largest no..
So There are infinitly many such numbers..
References :
Consider:
40 = 15 + 25
42 = 15 + 27
44 = 35 + 9
46 = 25 + 21
48 = 15 + 33
Each of these has as one of the terms of the addition a multiple of 5.
We can add any multiple of 10 to said multiple of 5 to get another
pair of composite odd numbers which has an even sum,
and which is 10 * n larger than these numbers.
So all numbers 40 and above can be written as the sum of two odd composites
(often in many different ways, but we only need this one).
Now consider 38:
The odd composite numbers < 38 are
9,15,21,25,27,33
and no two of them add up to 38.
So the answer is 38.
References :