The fact is, exceptions to the rule of "c < d" are hard to find, so while one can appreciate that the theorem is hard to PROVE, as a matter of experiment, one would suppose that the exceptions "peter out" pretty quickly.
Wikipedia gives the example of 2^7 = 3 + 5^3, with 128 > 30, as one of these exceptions. Of course, these are very small numbers. The next example in this pattern that I found was 2^14 = 759 + 5^6, with 16384 > 7590 ... hmmm. Well, and then 2^28 = 24294831 + 5^13, with 268435456 > 242948310 .
It takes a little work, because bc doesn't do factors, although the bash shell will take you pretty high. So we've got real power at our fingertips these days.
Well, they don’t peter out all that fast! Note this:
for( i=7 ; i<1000 ; i++ ){
for( j=1 ; 5^j < 2^i ; j++ ){}
if((2^i-5^(j-1))*10 < 2^i ) i
}
7
14
21
28
72
79
86
93
137
144
151
158
165
209
216
223
230
274
281
288
295
302
346
353
360
367
411
418
425
432
476
483
490
497
504
548
555
562
569
613
620
627
634
641
685
692
699
706
750
757
764
771
815
822
829
836
843
887
894
901
908
952
959
966
973
980
So these are the values of i such that (2^i - 5^j) *10 < 2^i, for some j. The last example has:
2^980-5^422
98572376703202965492169752825178002041673673337303057739076283487555\
46960879767539742261910099240198190716485086873048815196627230093618\
84287626693787764142921903239868072489333147534493860489512377512967\
02300928414227090926298660480965721138994950586311160850663090058942\
8048219799410664754551
2^980
10218702384817765435680628290748613458265350453429542612493041881278\
52488636909601681898478332229478957743332784226557564913888250057519\
95429845596072183368720384290455095586637697931337951384943751851865\
32064890845853749530218856391110938974453986086436459043203870933208\
875495579361330830770176
So recall that the theorem has that epsilon. It only says the “exceptions” succeed by less than an arbitrarily small exponential margin, after a certain limit.