[Guest]

What is the largest number that cannot be expressed as a sum of different square numbers?

(square numbers : 0,1,4,9,16,25,...)

[Guest]

Infinity

[Guest]

-1

[Guest]

the question is a bit ambiguous. how many square numbers are you allowed to use to get to the total?

if 2 or 3, then as far as I'm aware there is no number beyond which another bigger number doesn't exist.

if 4 however, I don't think there's any number you can't get to.

Edited August 5, 2010 by phillip1882

[Guest]

Completely misread the OP.

Edited August 5, 2010 by psychic_mind

[Guest]

the question is a bit ambiguous. how many square numbers are you allowed to use to get to the total?

if 2 or 3, then as far as I'm aware there is no number beyond which another bigger number doesn't exist.

if 4 however, I don't think there's any number you can't get to.

as many square numbers as you can use but each one should be different i.e. you cannot use twice

[Guest]

as many square numbers as you can use but each of them should be different,

for example;

136 =1+4+9+16+25+81

[Guest]

I think this is about combination. At some point, the number of combinations among the perfect squares will be much grater than the sum of all these squares. So even though some combinations might yield the same result (i.e 9 + 16 = 25, 4 + 9 + 36 = 49) there won't be any number that won't be covered after a certain value. What he wants to know is what that value is. I have no idea.

[Guest]

128

[Guest]

I think 128 is the largest number.

[unreality]

agreeing with phillip's request for clarification

Edited August 5, 2010 by unreality

[Guest]

agreeing with phillip's request for clarification

[/qu

clarification: We are looking for a number which is not expressed as sum of different square numbers;

for example 136 is no a number we are looking for because it can be expressed as 1+4+9+16+25+81 or 100+36 they are all square numbers,

but 43 is the number we may look for because it is not expressed as sum of some different square numbers.

question is what is the largest number satisfying this condition.

I agree to kewal the number should be 128 I think.

[Guest]

As the term square number is defined as a perfect square, the largest number that cannot be expressed as a sum of different square numbers is 128.

If, on the other hand, we allow any squared number, then there may be no number than cannot be expressed as the sum of different squared numbers. For example, 128 can be expressed as:

(8√2)² + 0²

or, with fractions

7.8² + 6² + 4.6² + 3² + 1²

or, with imaginary numbers (eliminating any negative integer as a solution)

8² + 7² + 4² + i²

[Quantum.Mechanic]

Anyone have a proof?