Is there a power of 666 such that its decimal notation starts with the digits 123456789?
One may replace "starts" by "ends" for a much easier challenge ;)
Below, L666 is the common log of 666; we continue adding it to Lp, which is the log of the power of 666 that we are up to. To begin with 123456789, the power must have a logarithm whose mantissa is within certain bounds of the mantissa of Lp. L123p has the mantissa which should not be exceeded; L123m has a mantissa that must be exceeded. Originally the lowering of this second mantissa was to allow for rounding errors; but the lowering is actually more than needed, but it does show a progression, apparently resulting from 666^40,894,182 beginning with a 1 followed by eight zeros and then nonzero digits.
10 point 7
20 L666=log(666)/log(10)
21 P=1:Lp=L666
30 L123=log(123456789)/log(10)
31 L123p=log(123456790)/log(10)8
32 L123m=log(1234567881)/log(10)9
35 while Found=0
40 P=P+1:Lp=Lp+L666
45 if Lpint(Lp)>L123m and Lpint(Lp)<L123p then
50 :print P;int(Lp);10^(Lpint(Lp))
90 wend
run
6722542 18980924 1.234567881198084806530809835259084
47616724 134444593 1.234567883508729726271308676128429
88510906 249908262 1.234567885819374650336462342734267
129405088 365371931 1.234567888130019578726270843170721
170299270 480835600 1.234567890440664511440734185531911
211193452 596299269 1.234567892751309448479852377911957
Break in 60
So the first power of 666 that has the required characteristics is represented by the 5th line above: 666^170,299,270 = 1.23456789044066451144073 x 10^480,835,600. (I've dropped the last 10 shown digits from the precision, as those are untrustworthy due to the 9digit power of 10 (characteristic).)

Posted by Charlie
on 20060831 13:51:05 