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e-Prime


An e-prime is a prime number appearing in the decimal expansion of e. The first few are 2, 271, 2718281, 2718281828459045235360287471352662497757247093699959574966967627724076630353547594571, ... (OEIS A007512). The numbers of digits in these examples are 1, 3, 7, 85, 1781, 2780, 112280, 155025, ... (OEIS A064118). The following table summarizes the largest known such primes.

ndiscoverer
2780E. W. Weisstein (Jan. 17, 2005)
112280E. W. Weisstein (Jul. 3, 2009)
155025E. W. Weisstein (Oct. 11, 2010)

Another set of e-related primes is the positive integers n such that |_e^n_| is prime, where |_x_| is the floor function. The first few are 1, 2, 18, 50, 127, 141, 267, 310, 2290, 4487, 5391, ... (OEIS A050808), corresponding to the primes 2, 7, 65659969, 5184705528587072464087, ... (OEIS A050809).

Similarly, the first few n such that [e^n] is prime, where [x] is the ceiling function are 1, 5, 7, 10, 105, ... (OEIS A059303), with no others less than 10^4, corresponding to the primes 3, 149, 1097, 22027, 3989519570547215850763757278730095398677254309, ... (OEIS A118840).

The first n-digit primes (excluding numbers with leading zeros) in the decimal expansion of e for n=1, 2, ... are 2, 71, 271, 4523, 74713, 904523, 2718281, 72407663, ... (OEIS A095935), which occur at positions 0, 1, 0, 14, 24, 12, 0, 64, 19, 99, 37, 53, ... (OEIS A115019), counting the leading 2 in the decimal expansion of e as position 0.


See also

Constant Primes, e, Integer Sequence Primes, Phi-Prime, Pi-Prime

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References

Pegg, E. Jr. and Weisstein, E. W. "Mathematica's Google Aptitude." MathWorld Headline News, Oct. 13, 2004. http://mathworld.wolfram.com/news/2004-10-13/google/.Pickover, C. A. "2, 271, 2718281" Ch. 95 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 198 and 333-334, 2002.Prime Curios! "2718281." http://primes.utm.edu/curios/page.php?number_id=1181.Sloane, N. J. A. Sequences A007512/M2184, A050808, A050809, A059303, A064118, A095935, A115019, and A118840 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

e-Prime

Cite this as:

Weisstein, Eric W. "e-Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/e-Prime.html

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