Euler's identity

Question

Warning: this question requires knowledge of complex numbers.

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An Euler's identity is an identity, in which each of the following appears once and only once:

• the constant $0$: neutral element for addition;
• the constant $1$: neutral element for multiplication;
• the constant $e$: base of natural logarithm;
• the constant $i$: square root of $-1$;
• the constant $\pi$: ratio of a circle's circumference to its diameter;
• the operation $+$: addition;
• the operation $\times$: multiplication;
• the operation $^\wedge$: exponentiation;
• the equal sign $=$: symbol for equality.

How many different Euler's identities exist?

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Note:

• The use of parentheses $()$ is unlimited.
• Other than the above mentioned symbols (including parentheses), no other symbol is allowed.
• Because of commutativity, $a + b$ and $b + a$ are considered the same. Same with $a \times b$ and $b \times a$.

Answer 1

Besides the famous identity, I've also found

12 more that are of a similar flavor, using the fact that any number to the 0th power, including complex numbers, is equal to 1: $$ e^{0 \times (i + \pi)} = 1 \\ i^{0 \times (\pi + e)} = 1 \\ \pi^{0 \times (e + i)} = 1 \\ (e + i)^{0 \times \pi} = 1 \\ (i + \pi)^{0 \times e} = 1 \\ (\pi + e)^{0 \times i} = 1 \\ (e + \pi \times i)^0 = 1 \\ (\pi + i \times e)^0 = 1 \\ (i + e \times \pi)^0 = 1 \\ ((e + \pi) \times i)^0 = 1 \\ ((\pi + i) \times e)^0 = 1 \\ ((i + e) \times \pi)^0 = 1 $$

I'll keep looking for more, but a proof that all of the possible identities are found will possibly require a computer search, which I don't know how to do.

Answer 2

I've only found the famous one:

$e^{i\pi}+1=0$

And possibly it's the only one due to:

The $e,\pi, i$ could only used once for each, and we must eliminate out those irrational numbers $e,\pi$ and complex number $i$ in the equation. So the possibility is to use the constants $0, 1$ with proper operations to eliminate them into integers.

Seems it's quite impossible that only 2 constants could be used but there are 3 irrational/complex numbers. So need to put $e,\pi, i$ together and only the famous one $e^{i\pi}+1=0$ found currently.