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High-Precision Fraud

A high-precision fraud is a false mathematical identity whose two sides agree numerically to so many digits that finite-precision computation appears to confirm it. Such deceptive agreement often occurs because the first nonzero term omitted from an approximation is extraordinarily small (Borwein and Borwein 1992).

For example, let

S=sum_(k=-infty)^infty1/(10^((k/100)^2))=theta_3(0,10^(-1/10000)),

where theta_3 is a Jacobi theta function. Then

S=100sqrt(pi/(ln10))+1.3809...×10^(-18613),

so the false identity obtained by omitting the error term agrees to more than 18000 decimal places (Borwein and Borwein 1992). High-precision frauds are closely related to almost integers and near-identities.

See also

Almost Integer, Identity, Series

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. "High Precision Fraud." §1.4 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 11-15, 2004.Borwein, J. M. and Borwein, P. B. "Strange Series and High Precision Fraud." Amer. Math. Monthly 99, 622-640, 1992.

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High-Precision Fraud

Cite this as:

Weisstein, Eric W. "High-Precision Fraud." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/High-PrecisionFraud.html

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