The Enigma of Cleo Vladimir Reshetnikov

Vladimir Reshetnikov (born 1979) is a mathematician and software developer of Uzbekistani-American origin. He gained internet notoriety under the pseudonym Cleo for his uncanny ability to solve highly complex mathematical integration problems on the Stack Exchange platform.

Quick Reference: Profile Summary

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The Mystery of the "Mathematical Genius"

Between late 2013 and late 2015, a user known as Cleo became a focal point of discussion within the mathematical community. The account was famous for providing perfectly accurate solutions to advanced integrals that had baffled other experts, yet they did so without providing any intermediate steps or proofs.

Key characteristics of Cleo's activity included:

• Extreme Speed: Solutions often appeared only a few hours after a problem was posted.
• Lack of Methodology: No explanations were provided, only the final result.
• High Precision: The answers were consistently correct.

Because of this, a debate erupted online: was Cleo a singular, transcendent genius, or was the account a front for a collective of mathematicians? Some observers drew parallels between Cleo and the legendary Srinivasa Ramanujan, who similarly produced profound mathematical results without conventional proofs.

The Persona

The account maintained a very low profile. In their bio, the user stated:

"My real name is Cleo, I'm female. I have a medical condition that makes it very difficult for me to engage in conversations, or post long answers, sorry for that. I like math and do my best to be useful at this site, although I realize my answers might be not useful for everyone."

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The Definitive Case: A Complex Integral

The most famous instance of Cleo's prowess occurred on November 11, 2013. A user named Laila Podlesny posted a daunting integral:

$I=\int _{-1}^{1}{\frac {1}{x}}{\sqrt {\frac {1+x}{1-x}}}\ln \left({\frac {2x^{2}+2x+1}{2x^{2}-2x+1}}\right)\ \mathrm {d} x$

The challenge was immense because:

• Mathematica could not find a closed-form expression.
• Maple failed to solve it.
• WolframAlpha and ISC+ returned no plausible candidates.
• The approximate numeric value was known to be: $I=8.372211626601275661625747121\ldots$

The Resolution

Only four and a half hours later, Cleo provided the solution:

$I=4\pi \operatorname {arccot} {\sqrt {\varphi }}$

Where $\varphi$ (the golden ratio) is defined as:

While the answer was correct, it sparked controversy. The Stack Exchange community argued that providing answers without proofs defeated the educational purpose of the site. However, two days later, Ron Gordon (a former physicist and patent agent) published a rigorous proof. Gordon used a keyhole contour integral and symmetry analysis to reduce an eighth-degree polynomial into a quadratic equation, confirming Cleo's result. Gordon's work was so highly regarded that he was later called the "Master of Integration" on the /r/math subreddit.

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Unmasking the Identity

For years, the identity of Cleo remained a mystery. However, in January 2025, internet investigators successfully tracked the account back to its true owner.

The Reveal: Cleo was not a woman with a medical condition, but rather Vladimir Reshetnikov.

The transition of the identity can be viewed as: Cleo $\rightarrow$ Anonymous Genius $\rightarrow$ Vladimir Reshetnikov