Special Numbers
Mathematics is a rich and varied landscape, full of mathematical flora and fauna. Finding your way around can be tricky at the best of times, so it’s good to have some pointers to give you a hint on which way to go next.
Of all mathematical ideas, numbers are among the simplest, so let’s look at some special numbers. I have made an Excel version of the table below, together with some other bits and pieces.
Hmm – 153, 370, 371 and 407 are calling out that all numbers are special, and so indeed all of them are (for example, these numbers are the only four with which property? See below for solution). But still…
Indeed, the numbers in the table below do appear much more often in problems (both in and out of school). Perhaps 5% to 10% of problems use such numbers, and so will be easier to solve if you know why they are special.
For example, if you see 343, almost certainly it is part of the problem because it is 7x7x7. In contrast, I don’t remember ever seeing 342 in a problem. Numbers like 342 may appear very occasionally, but for no particular reason – and therefore much, much less often.
#  Powers  Exponentials  Factorials*  Primes  
n  n^{2}  n^{3}  2^{n}  3^{n}  5^{n}  n!  p_{n} 
0 
0 
0 
1 
1 
1 
1  
1 
1 
1 
2 
3 
5 
1 
2 
2 
4 
8 
4 
9 
25 
2 
3 
3 
9 
27 
8 
27 
125 
6 
5 
4 
16 
64 
16 
81 
625 
24 
7 
5 
25 
125 
32 
243 
3125 
120 
11 
6 
36 
216 
64 
729 
15625 
720 
13 
7 
49 
343 
128 
2187 
78125 
5040 
17 
8 
64 
512 
256 
6561 
40320 
19 

9 
81 
729 
512 
19683 
362880 
23 

10 
100 
1000 
1024 
59049 
3628800 
29 

11 
121 
1331 
2048 
31 

12 
144 
1728 
4096 
37 

13 
169 
2197 
8192 
41 

14 
196 
2744 
16384 
43 

15 
225 
3375 
32768 
47 

16 
256 
4096 
65536 
53 

17 
289 
4913 
131072 
59 

18 
324 
5832 
262144 
61 

19 
361 
6859 
524288 
67 

20 
400 
8000 
1048576 
71 

21 
441 
73 

22 
484 
79 

23 
529 
83 

24 
576 
89 

25 
625 
97 

26 
676 

27 
729 

28 
784 
— — Legend — — 

29 
841 
Red = must know 

30 
900 
Black = should know 

31 
961 
Blue – nice to know 

32 
1024 
Green – just for fun 
* Note on factorials: 4! = 4x3x2x1 = 24 5! = 5x4x3x2x1 = 120 6! = 6x5x4x3x2x1 = 720 etc
I originally made this spreadsheet for upper secondary student use, but even those who are younger (or older!) readers would benefit from familiarity with the numbers in this table.
153, 370, 371, 407 ?
153 = 1^{3} + 5^{3} + 3^{3}
370 = 3^{3} + 7^{3} + 0^{3}
371 = 3^{3} + 7^{3} + 1^{3}
407 = 4^{3} + 0^{3} + 7^{3}
Related articles
 Strategies For ‘Attacking’ Maths Problems, A Guide For Students And Teachers (tutoringtoexcellence.blogspot.com)
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