

A060295


Decimal expansion of exp(Pi*sqrt(163)).


27



2, 6, 2, 5, 3, 7, 4, 1, 2, 6, 4, 0, 7, 6, 8, 7, 4, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 5, 0, 0, 7, 2, 5, 9, 7, 1, 9, 8, 1, 8, 5, 6, 8, 8, 8, 7, 9, 3, 5, 3, 8, 5, 6, 3, 3, 7, 3, 3, 6, 9, 9, 0, 8, 6, 2, 7, 0, 7, 5, 3, 7, 4, 1, 0, 3, 7, 8, 2, 1, 0, 6, 4, 7, 9, 1, 0, 1, 1, 8, 6, 0, 7, 3, 1, 2, 9, 5, 1, 1, 8, 1
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OFFSET

18,1


COMMENTS

One could observe that the last four of Class Number 1 expressions in T. Piezas "Ramanujan Pages" could be expressed as the following approximation:
exp(Pi*sqrt(19+24*n)) =~ (24*k)^3 + 31*24
which gives 4 (four) "almost integer" solutions:
1) n = 0, 19+24*0 = 19, k = 4;
2) n = 1, 19+24*1 = 43, k = 40;
3) n = 2, 19+24*2 = 67, k = 220;
4) n = 6, 19+24*6 = 163, k = 26680; this of course is the case for Ramanujan constant vs. its integer counterpart approximation. (End)
Also if one expands the left part above to exp(Pi*sqrt(b(n))) where b(n) = {19, 25, 43, 58, 67, 163, 232, ...} then the expression (exp(Pi*sqrt(b(n))))/m (where m is either integer 1 or 8) yields values being very close to whole integer value:
Note, that the first differences of b(n) are all divisible by 3, giving after the division: {2, 6, 5, 3, 32, 33, ...}. (End)
This constant was discovered by Hermite (1859).
It is sometimes called "Ramanujan's constant" due to an April Fool's joke by Gardner (1975) in which he claimed that Ramanujan conjectured that this constant is an integer, and that a fictitious "John Brillo" of the University of Arizona proved it on May 1974.
In fact, Ramanujan studied similar nearintegers of the form exp(Pi*sqrt(k)) (e.g., A169624), but not this constant.
Gauld (1984) discovered that (Pi*sqrt(163))^e = 22806.9992... is also a nearinteger. (End)


REFERENCES

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, p. 106.
Harold M. Stark, An Introduction to Number Theory, Markham, Chicago, 1970, p. 179.


LINKS

David Barry Gauld, Problem 12 revisited, New Zealand Mathematical Society Newsletter 32 (December 1984), p. 17.


FORMULA



EXAMPLE

The Ramanujan number = 262537412640768743.99999999999925007259719818568887935...


MATHEMATICA

RealDigits[N[E^(Pi*Sqrt[163]), 110]][[1]]


PROG

(PARI) default(realprecision, 20080); x=exp(Pi*sqrt(163))/10^17; for (n=18, 20000, d=floor(x); x=(xd)*10; write("b060295.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009
(Magma) R:= RealField(); Exp(Pi(R)*Sqrt(163)) // G. C. Greubel, Feb 15 2018


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KEYWORD



AUTHOR



STATUS

approved



