Conference rusure::math

    I don't think so.  Here's why:
    
    Assume an integer X such that   X=e^(pi*163^.5).     Then,
                                X*e^i=e^(pi*163^.5)*e^i
    				X*e^i=e^(pi*i)*e^(163^.5)
                  X*[cos(1)+i*sin(1)]=-e^(163^.5)
    
    Since the lhs of the equation has a non-zero imaginary part and
    the rhs does not, there can be no X that would satisfy the initial
    assumption.
    

Re .1: the third step doesn't follow from the second; i.e.

		e^x * e^y  is not e^(x*y), its e^(x+y)


VAX FORTRAN (using REAL*16) believes the answer is no.



I believe its infinitely unlikely the answer is yes.....

    Re .2:
    
    REAL*16 _might_ be accurate enough to tell, if you are sure the error
    of the computation is confined to the last 12 or 13 bits of the 113-bit
    fraction of the real number representation.
    
    
    				-- edp 

As I recall, this problem or one very like it appeared in an April 1 issue of
Martin Gardner's 'Math. Games' column several years back; there it was
'announced' that someone had finally 'proved' it to be an integer (along with
several other mathematical practical jokes).

MAPLE yields 
	evalf(exp(Pi*sqrt(163)),40) =
		262537412640768743.9999999999992500725944

of which the last couple of digits are probably not trustworthy.

Lynn Yarbrough 

    Ooops! is right.  Thanks, Richy - it seemed too simple.
    
    						/Dwayne
    

FORTRAN REAL*16 yielded a result close to MAPLE's - the number of 9's
and the first non-9 digit were the same, anyway.....

    The Gelfond-Schneider theorem states that if a and b are algebraic
    numbers where a .ne. 0 or 1, and b irrational,then a^b is
    guaranteed transcendental.  This can be used to prove that this
    expression is not an integer.  See, I think, Harold Stark's
    'Introduction to Nubmer Theory', and also a note somewhere early
    in this conference on the "MP" mulitprecision package - this is one
    of the test cases, and the ACM TOMS paper on the package mentions
    some other references on this equation which explain why it comes
    so close to an integer.

    - Jim

I've long been impressed with the simple proof that all noninteger square
roots of integers aren't ever rational.

Perhaps there's an almost-as-simple proof that shows that for all positive
integers n,

	x^(y*n)

is always irrational as long as x and y are both real unrelated irrationals
themselves.  But how do we define "unrelated" ?

I only have a feel for unrelatedness in this context.  For example, in
our example, I suspect

	e^(pi*n^.5)

is always irrational.

/Eric

Is 163 the integer n that causes

	v = e^(pi*n^.5)

to be closest to an integer ?

I think that by "closest" I mean that

	abs(v-int(v+.5)
	--------------
	      n

is closest to 0.

/Eric

re .0:
	I believe this result (actually, whole classes of such near-integers)
	are due to the famous Indian mathematician, Srinivasa Ramanujan.
	The Sci.Am. article attributed it to this amazing self-taught
	mathematician.

re .8:
	The Gelfond-Schneider theorem doesn't apply -- neither e nor
	pi*sqrt(n) is an algebraic number.

I fail to see the significance of near-integers.  I guess, not being
very expert on infinities, that there must be an infinity of irrational
near-integer numbers.  Heck, the places integers are are kind of
accidental anyway, and there's only an infinity of them, but in betwee
consecutive integers is an infinity of non-integers.

Why aren't people interest in numbers that are nearly but not quite
4.55423456234234636234686858909181273657286965322498?

Tom

Can someone find a few positive integers k such that e^k is within 10^-6
of an integer?  Assuming that the fractional part of e^k is randomly
distributed in the range 0 to 1 (exclusive), roughly 2 out of every 10^6 
choices for k get this close.

For e^(pi*sqrt(k)), we'd expect fewer than 1 in 10^12 values of k to be
closer to an integer than is k=163.  Isn't it surprising that such a
small k gets so close?  Is there a pattern in such k?

    Re .13:
    
    > For e^(pi*sqrt(k)), we'd expect fewer than 1 in 10^12 values of k to
    > be closer to an integer than is k=163.  Isn't it surprising that such a
    > small k gets so close?
    
    Yes, especially since e^(pi*sqrt(58)) is within 2*10^-7 of an integer. 
    
    
    				-- edp

    k	first digits of fractional part of e^(pi*sqrt(k))
    6	.9908
    17	.9976
    18	.9929
    22	.9981
    25	.9996
    
    Looks suspicious to me.  Could somebody with an extended-precision
    product supply more?
    
    
    				-- edp 

Below are all of the near-integers (first fractional digit = 0 or 9) from 1
to 200. (Thanks to MAPLE.)

One reason that these are interesting is the concept of 'easy' and 'hard'
integers, which depends on how compactly they can be described. An integer 
is easy if there is a functional way to describe it that requires fewer 
symbols than log(10) of the number, hard if not. 1000000000 is easy,
1010010009 is hard. 24591257752 is 'nearly easy' in spite of its apparent 
randomness, since it is nearly exp(Pi*sqrt(58)).

2, 85.019695223207217582510872858830
6, 2197.9908695437080831653650130144
7, 4071.9320952252610985245683325575
13, 83047.947369666838051322861479741
17, 422150.99767568045162231182799352
18, 614551.99288561963541392982762989
22, 2508951.9982574244671655291941215
25, 6635623.9993411342332662640670987
27, 12288743.983978651422664663257042
28, 16580630.988125584818725036441232
30, 29716459.097399582162171866835762
31, 39493537.906571037292282005299380
33, 68824876.077757883360926406578785
37, 199148647.99997804655185676650091
42, 695295413.03013616006103923971391
43, 884736743.99977746603490666193742
56, 16220256788.059490559297281550256
58, 24591257751.999999822213241469575
67, 147197952743.99999866245422450681
74, 545518122089.99917467885354985657
84, 3196802717869.9132927417022131627
88, 6294840129559.9232665243470929326
94, 16909062792429.904032549249131633
96, 23340144267268.053501933178112493
100, 44031505860632.029011400544566529
103, 70292286279654.001941288875880775
125, 1795524644499992.9496913404329661
127, 2375421230612382.9027861115219042
129, 3135714579766712.9603161716718752
136, 8151279246142431.9199301641981361
140, 13916138006676652.068427231284784
148, 39660184000219160.000966674358570
149, 45116546012289599.991830287000358
163, 262537412640768743.99999999999921
164, 296853791705948489.00267262483543
171, 694282057876537408.01228868670799
172, 782759106183327768.04858787037421
174, 993940015039984910.94791838785323
177, 1418556986635586485.9961793552496
182, 2549431380498057734.0336489122058
183, 2863790268421473279.9531058716544
187, 4545336381788161590.0735791051683
188, 5097838276117546947.0732677708421