If primes are like diamonds, powers of 2 are like talc. Primes don’t crumble under division, because they can’t be divided by any number but themselves and one. Powers of 2 crumble more than any other numbers. The contrast is particularly strong when the primes are Mersenne primes, or equal to a power of 2 minus 1:

3 = 4-1 = 2^2 – 1.

4, 2, 1.

7 = 8-1 = 2^3 – 1.

8, 4, 2, 1.

31 = 32-1 = 2^5 – 1.

32, 16, 8, 4, 2, 1.

127 = 2^7 – 1.

128, 64, 32, 16, 8, 4, 2, 1.

8191 = 2^13 – 1.

8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

131071 = 2^17 – 1.

131072, 65536, 32768, 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

524287 = 2^19 – 1.

524288, 262144, 131072, 65536, 32768, 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

2147483647 = 2^31 – 1.

2147483648, 1073741824, 536870912, 268435456, 134217728, 67108864, 33554432, 16777216, 8388608, 4194304, 2097152, 1048576, 524288, 262144, 131072, 65536, 32768, 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.

Are Mersenne primes infinite? If they are, then there will be just as many Mersenne primes as powers of 2, even though very few powers of 2 create a Mersenne prime. That’s one of the paradoxes of infinity: an infinite part is equal to an infinite whole.

But are they infinite? No-one knows, though some of the greatest mathematicians in history have tried to find a proof or disproof of the conjecture. A simpler question about powers of 2 is this: Does every integer appear as part of a power of 2? I can’t find one that doesn’t:

0 is in 1024 = 2^10.

1 is in 16 = 2^4.

2 is in 32 = 2^5.

3 is in 32 = 2^5.

4 = 2^2.

5 is in 256 = 2^8.

6 is in 16 = 2^4.

7 is in 32768 = 2^15.

8 = 2^3.

9 is in 4096 = 2^12.

10 is in 1024 = 2^10.

11 is in 1099511627776 = 2^40.

12 is in 128 = 2^7.

13 is in 131072 = 2^17.

14 is in 262144 = 2^18.

15 is in 2097152 = 2^21.

16 = 2^4.

17 is in 134217728 = 2^27.

18 is in 1073741824 = 2^30.

19 is in 8192 = 2^13.

20 is in 2048 = 2^11.

666 is in 182687704666362864775460604089535377456991567872 = 2^157.

1066 is in 43556142965880123323311949751266331066368 = 2^135.

1492 is in 356811923176489970264571492362373784095686656 = 2^148.

2014 is in 3705346855594118253554271520278013051304639509300498049262642688253220148477952 = 2^261.

I’ve tested much higher than that, but testing is no good: where’s a proof? I don’t have one, though I conjecture that all integers do appear as part or whole of a power of 2. Nor do I have a proof for another conjecture: that all integers appear infinitely often as part or whole of powers of 2. Or indeed, of powers of 3, 4, 5 or any other number except powers of 10.

I conjecture that this would apply in all bases too: In any base b all n appear infinitely often as part or whole of powers of any number except those equal to a power of b.

1 is in 11 = 2^2 in base 3.

2 is in 22 = 2^3 in base 3.

10 is in 1012 = 2^5 in base 3.

11 = 2^2 in base 3.

12 is in 121 = 2^4 in base 3.

20 is in 11202 = 2^7 in base 3.

21 is in 121 = 2^4 in base 3.

22 = 2^3 in base 3.

100 is in 100111 = 2^8 in base 3.

101 is in 1012 = 2^5 in base 3.

102 is in 2210212 = 2^11 in base 3.

110 is in 1101221 = 2^10 in base 3.

111 is in 100111 = 2^8 in base 3.

112 is in 11202 = 2^7 in base 3.

120 is in 11202 = 2^7 in base 3.

121 = 2^4 in base 3.

122 is in 1101221 = 2^10 in base 3.

200 is in 200222 = 2^9 in base 3.

201 is in 12121201 = 2^12 in base 3.

202 is in 11202 = 2^7 in base 3.

1 is in 13 = 2^3 in base 5.

2 is in 112 = 2^5 in base 5.

3 is in 13 = 2^3 in base 5.

4 = 2^2 in base 5.

10 is in 1003 = 2^7 in base 5.

11 is in 112 = 2^5 in base 5.

12 is in 112 = 2^5 in base 5.

13 = 2^3 in base 5.

14 is in 31143 = 2^11 in base 5.

20 is in 2011 = 2^8 in base 5.

21 is in 4044121 = 2^16 in base 5.

22 is in 224 = 2^6 in base 5.

23 is in 112341 = 2^12 in base 5.

24 is in 224 = 2^6 in base 5.

30 is in 13044 = 2^10 in base 5.

31 = 2^4 in base 5.

32 is in 230232 = 2^13 in base 5.

33 is in 2022033 = 2^15 in base 5.

34 is in 112341 = 2^12 in base 5.

40 is in 4022 = 2^9 in base 5.

1 is in 12 = 2^3 in base 6.

2 is in 12 = 2^3 in base 6.

3 is in 332 = 2^7 in base 6.

4 = 2^2 in base 6.

5 is in 52 = 2^5 in base 6.

10 is in 1104 = 2^8 in base 6.

11 is in 1104 = 2^8 in base 6.

12 = 2^3 in base 6.

13 is in 13252 = 2^11 in base 6.

14 is in 144 = 2^6 in base 6.

15 is in 101532 = 2^13 in base 6.

20 is in 203504 = 2^14 in base 6.

21 is in 2212 = 2^9 in base 6.

22 is in 2212 = 2^9 in base 6.

23 is in 1223224 = 2^16 in base 6.

24 = 2^4 in base 6.

25 is in 13252 = 2^11 in base 6.

30 is in 30544 = 2^12 in base 6.

31 is in 15123132 = 2^19 in base 6.

32 is in 332 = 2^7 in base 6.

1 is in 11 = 2^3 in base 7.

2 is in 22 = 2^4 in base 7.

3 is in 1331 = 2^9 in base 7.

4 = 2^2 in base 7.

5 is in 514 = 2^8 in base 7.

6 is in 2662 = 2^10 in base 7.

10 is in 1054064 = 2^17 in base 7.

11 = 2^3 in base 7.

12 is in 121 = 2^6 in base 7.

13 is in 1331 = 2^9 in base 7.

14 is in 514 = 2^8 in base 7.

15 is in 35415440431 = 2^30 in base 7.

16 is in 164351 = 2^15 in base 7.

20 is in 362032 = 2^16 in base 7.

21 is in 121 = 2^6 in base 7.

22 = 2^4 in base 7.

23 is in 4312352 = 2^19 in base 7.

24 is in 242 = 2^7 in base 7.

25 is in 11625034 = 2^20 in base 7.

26 is in 2662 = 2^10 in base 7.

1 is in 17 = 2^4 in base 9.

2 is in 152 = 2^7 in base 9.

3 is in 35 = 2^5 in base 9.

4 = 2^2 in base 9.

5 is in 35 = 2^5 in base 9.

6 is in 628 = 2^9 in base 9.

7 is in 17 = 2^4 in base 9.

8 = 2^3 in base 9.

10 is in 108807 = 2^16 in base 9.

11 is in 34511011 = 2^24 in base 9.

12 is in 12212 = 2^13 in base 9.

13 is in 1357 = 2^10 in base 9.

14 is in 314 = 2^8 in base 9.

15 is in 152 = 2^7 in base 9.

16 is in 878162 = 2^19 in base 9.

17 = 2^4 in base 9.

18 is in 218715 = 2^17 in base 9.

20 is in 70122022 = 2^25 in base 9.

21 is in 12212 = 2^13 in base 9.

22 is in 12212 = 2^13 in base 9.