Verify that ## A(225)\geq 1 ##

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In summary, the given conversation discusses the divisors of 225 and the real-valued nonprincipal characters modulo 16. It provides a table of the values for these characters and states that the real-valued characters are ##\chi_1,\chi_2,\chi_5,\chi_6##. It also mentions a calculation for ##A(225)##, which evaluates to 3 for ##\chi_2, \chi_5,## and ##\chi_6##.
  • #1
Math100
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Homework Statement
For each real-valued nonprincipal character ## \chi\pmod {16} ##, verify that ## A(225)\geq 1 ##.
Relevant Equations
Let ## \chi ## be any real-valued character mod ## k ## and let ## A(n)=\sum_{d\mid n}\chi(d) ##. Then ## A(n)\geq 0 ## for all ## n ##, and ## A(n)\geq 1 ## if ## n ## is a square.
Let ## n=225 ## and ## d ## be the divisors of ## n ##.
Then ## d=\left \{ 1, 3, 5, 9, 15, 25, 45, 75, 225 \right \} ##.
Note that the real-valued nonprincipal characters ## \chi\pmod {16} ## are ## \chi(1), \chi(7), \chi(9), \chi(15) ##.
Observe that ## \chi(1)=1, \chi(7)=\pm 1, \chi(9)=\pm 1, \chi(15)=\pm 1 ##.
Thus ## A(225)=\sum_{d\mid 225}\chi(d)=\chi(1)+\chi(9)+\chi(15)=1+1+1=3\geq 1 ##.
Therefore, ## A(225)\geq 1 ##.

\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline n & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\
\hline \chi_{1}(n) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\hline \chi_{2}(n) & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\
\hline \chi_{3}(n) & 1 & i & i & 1 & -1 & -i & -i & -1 \\
\hline \chi_{4}(n) & 1 & -i & i & -1 & -1 & i & -i & 1 \\
\hline \chi_{5}(n) & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\
\hline \chi_{6}(n) & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\
\hline \chi_{7}(n) & 1 & i & -i & -1 & -1 & -i & i & 1 \\
\hline \chi_{8}(n) & 1 & -i & -i & 1 & -1 & i & i & -1 \\
\hline
\end{array}
 
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  • #2
The real valued characters are ##\chi_1,\chi_2,\chi_5,\chi_6## where ##\chi_1## is the principal character. I think you left out a calculation step which is confusing here. It should be ...
\begin{align*}
A(225)&=\sum_{d|225}\chi_2(d)=2 [\chi_2(1)+\chi_2(3)+\chi_2(9)]+\chi_2(5)+\chi_2(13)+\chi_2(15)\\
&=2[1-1+\chi_2(9)]+\chi_2(5)+1-1\\
&=2\chi_2(9)+\chi_2(5)=3
\end{align*}
and the same for ##\chi_5## and ##\chi_6.##
 
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FAQ: Verify that ## A(225)\geq 1 ##

What does "Verify that A(225)≥ 1" mean?

This statement is asking you to check whether the value of A at 225 is greater than or equal to 1. In other words, you need to evaluate the function A at the input value of 225 and see if the output is equal to or greater than 1.

How do I verify that A(225)≥ 1?

To verify this statement, you will need to have the function A and its corresponding equation or table of values. Plug in the input value of 225 into the function and calculate the output. If the output is equal to or greater than 1, then the statement is true.

Why is it important to verify this statement?

Verifying this statement helps to ensure the accuracy and validity of the function A. It allows you to check if the function is behaving as expected and if there are any errors or inconsistencies in the function.

Can I use a calculator to verify this statement?

Yes, you can use a calculator to evaluate the function A(225) and check if it is greater than or equal to 1. However, it is important to note that a calculator may not always be accurate and it is always best to double check your calculations.

What should I do if A(225) is not greater than or equal to 1?

If A(225) is not greater than or equal to 1, then the statement is false. This could mean that there is an error in the function or that the input value of 225 is not within the domain of the function. In this case, you may need to reevaluate the function or check the validity of the input value.

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