Integer tuples with equal L1 and L2 norms

Thread starter boy_travels
Start date Mar 27, 2009
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Integer L2

In summary, an integer tuple with equal L1 and L2 norms is a set of integers where the sum of their absolute values is equal to the square root of the sum of their squares. It has real-world applications in fields such as signal processing and cryptography. The calculation involves finding solutions to the equation x^2 + y^2 = z^2, and there is no limit to the number of integers in the tuple. Negative values are allowed, but the total sum of the absolute values must still be equal to the square root of the sum of their squares.

Mar 27, 2009

boy_travels

Let x and y be n-tuples of non-negative integers.

Furthermore,

sum x_i = sum y_i

and,

sum x_i^2 = sum y_i^2

Is it true that x must be a permutation of y?

Cheers!

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Mar 27, 2009

boy_travels

Sorry. I found a way to generate trivial counterexamples.

E.g. x = [16 13 9 4]
y = [17 12 8 5]

etc.

Related to Integer tuples with equal L1 and L2 norms

1. What is an integer tuple with equal L1 and L2 norms?

An integer tuple with equal L1 and L2 norms is a set of integers where the sum of their absolute values (L1 norm) is equal to the square root of the sum of their squares (L2 norm). This means that the distance of the tuple from the origin in both the L1 and L2 norm spaces is the same.

2. Are there any real-world applications for integer tuples with equal L1 and L2 norms?

Yes, integer tuples with equal L1 and L2 norms have various applications in fields such as signal processing, image compression, and data analysis. They can also be used in cryptography to generate secure keys.

3. How are integer tuples with equal L1 and L2 norms calculated?

The calculation of integer tuples with equal L1 and L2 norms involves finding the integer solutions to the equation x² + y² = z² where x and y are the integers in the tuple and z is the distance from the origin in the L1 and L2 norm spaces. This can be done using various mathematical methods, such as the Pythagorean triple formula.

4. Is there a limit to the number of integers in an integer tuple with equal L1 and L2 norms?

No, there is no limit to the number of integers in an integer tuple with equal L1 and L2 norms. However, as the number of integers increases, the difficulty of finding solutions to the equation x² + y² = z² also increases.

5. Can integer tuples with equal L1 and L2 norms have negative values?

Yes, integer tuples with equal L1 and L2 norms can have negative values. In fact, having negative values in the tuple can make it easier to find solutions to the equation x² + y² = z². However, the total sum of the absolute values of the integers in the tuple must still be equal to the square root of the sum of their squares.