What is the remainder?

Computer Science Level 3

${\color{#D61F06}\text{This problem's question:}}$ What is the remainder? $(2^{2^{2^{2^{2^{2^{2^{2^{2^2}}}}}}}} \bmod 18446744073709551619)$

The modulus is $2^{2^6}+3$ .

${\color{#EC7300}\text{As expressed above, this problem breaks Wolfram Mathematica 12.}}$ There exists another, equivalent Wolfram Mathematica 12 expression that works.

Hint: since no one else has solved the problem, use the PowerMod function and divide the power tower in half. That is the other, equivalent Wolfram Mathematica 12 expression that works.

A number theoretic approach may work.

The answer is 2047291311358963671.

1 solution

A Former Brilliant Member
Jul 21, 2019

The equivalent Wolfram Mathematica 12 expression is PowerMod[2^2^2^2^2, 2^2^2^2^2, 2^2^6 + 3] or $(\left(2^{2^{2^{2^2}}}\right)^{2^{2^{2^{2^2}}}} \bmod \left(2^{2^6}+3\right)) \Rightarrow 2047291311358963671$ It executes in a small number of milliseconds on my computer.

The PowerMod function uses the Mod function on intermediate values as needed to expedite the computation while working to the final answer. This function is used in cryptographic number theory.

$\log _{10}\left(2^{2^{2^{2^2}}} \log _{10}\left(2^{2^{2^{2^2}}}\right)\right) \approx 19732.5968855376$ .

What is the remainder?

The answer is 2047291311358963671.

1 solution

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