1. ## Number problem

P is the product and S the sum of two natural numbers different from zero, prove that there are infinite natural numbers that can NOT be obtained with P + S.

Example

5 can be obtained because: (1 * 2) + (1 +2) = 5

Not 6 .. (can test)

The problem is to show that there are infinite numbers such as 6.

2. Originally Posted by streethot
P is the product and S the sum of two natural numbers different from zero, prove that there are infinite natural numbers that can NOT be obtained with P + S.

Example

5 can be obtained because: (1 * 2) + (1 +2) = 5

Not 6 .. (can test)

The problem is to show that there are infinite numbers such as 6.

Let p be a natural number for which there exist two natural numbers s.t.

p = nm + n + m = (n+1)(m+1) - 1 ==> p - 1 = (n+1)(m+1).

If every natural number p can be obtained, then since every natural number can be written as p-1 we get that every natural number can be expressed as (n+1)(m+1)...but there are infinite very easy to find natural numbers that can NOT be expressed this way...

Tonio

3. Originally Posted by tonio
Let p be a natural number for which there exist two natural numbers s.t.

p = nm + n + m = (n+1)(m+1) - 1 ==> p - 1 = (n+1)(m+1).

If every natural number p can be obtained, then since every natural number can be written as p-1 we get that every natural number can be expressed as (n+1)(m+1)...but there are infinite very easy to find natural numbers that can NOT be expressed this way...

Tonio
I think Tonio meant p + 1 = (n+1)(m+1)

4. Originally Posted by aman_cc
I think Tonio meant p + 1 = (n+1)(m+1)
Of course. Thanx

Tonio

5. Originally Posted by tonio
Of course. Thanx

Tonio
@Tonio - Hi - I was trying this question as well. I rested it on the argument that if N can be expressed as mn+m+n => N+1 can't be expressed as such.

Is this true? (Initially it looked trivial to me, but am lost now)

6. Sorry I was completely wrong. Please ignore.