The Long Run Guess

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.oN

[quiznos00]

Is it of the form a * i, where a is an irrational number?

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Assuming you mean a is an irrational real number (because otherwise it's a repeat of your previous question), no.

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Hint: the number is of the form a + bi, where (a² + b²)^1/2 is an integer.

[_H_]

r exp(iφ), r∈ℕ, φ∈ℝ
still an infinite amount of possibilities for r as well as φ

I'll anyhow take a shot in the dark:
r∈{5,10}, φ∈{±atan(3/4),±atan(4/3)}

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No and no.

[_H_]

If abs(a),abs(b) ≤ 100 there are some more simple possibilities:

φ∈{±atan(x^±1):x∈{5/12, 7/24, 8/15, 9/40, 11/60, 12/35, 13/84, 16/63, 20/21, 20/99, 28/45, 33/56, 36/77, 39/80, 48/55, 60/91, 65/72}}

(of course these can lead to a, b way bigger than 100, but not every possibility has to be covered above 100) if φ is not an element of this set, either a or b are irrational (since you wrote in a previous answer, b isn't the irrational part, a is irrational), or you constructed the numbers some other way, but without knowing the algebraic curve you've chosen to do so, this wouldn't help anyhow, however as they are constructed to have this form in either way, it would be basically impossible to guess them...

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Is it related to Pi?

Nope.

This may have been before i understood the polar form of complex numbers. π does show up in the number if written in the form re^iθ. Apologies for any confusion.

abs(a),abs(b) ≤ 100

Yes.

φ∈{±atan(x^±1):x∈{5/12, 7/24, 8/15, 9/40, 11/60, 12/35, 13/84, 16/63, 20/21, 20/99, 28/45, 33/56, 36/77, 39/80, 48/55, 60/91, 65/72}}

φ is none of those numbers.

(since you wrote in a previous answer, b isn't the irrational part, a is irrational)

I said the number is not of the form a*i, where a is rational or real. That means, in a+bi, where a,b∈ℝ, one or both of a and b is irrational.

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Recap:

• The number is algebraic and irrational.
  
• The number is not real or imaginary (i.e. in rectangular form [a+bi, a∈ℝ, b∈ℝ], a and b are both non-zero).
  
• The number, written in polar form (re^iθ, r∈ℝ, 0≤θ<2π), contains π.
  
• r∈ℤ, r ≠ 5, r ≠ 10, |a| ≤ 100, |b| ≤ 100.
  
• tan(|θ|) ∉ A and cot(|θ|) ∉ A, where A = {3/4, 5/12, 7/24, 8/15, 9/40, 11/60, 12/35, 13/84, 16/63, 20/21, 20/99, 28/45, 33/56, 36/77, 39/80, 48/55, 60/91, 65/72}.
  

New hint: the number is a root of a non-zero
polynomial in one variable
with integer coefficients and
degree no greater than 5.