Tag Archives: bases

Talcum Power

by in Base Behavior, Mathematics, Powers, Prime numbers and tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment

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.

Factory Records

by in Base Behavior, Mathematics, Palindromic numbers, Prime numbers and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

The factors of n are those numbers that divide n without remainder. So the factors of 6 are 1, 2, 3 and 6. If the function s(n) is defined as “the sum of the factors of n, excluding n”, then s(6) = 1 + 2 + 3 = 6. This makes 6 a perfect number: its factors re-create it. 28 is another perfect number. The factors of 28 are 1, 2, 4, 7, 14 and 28, so s(28) = 1 + 2 + 4 + 7 + 14 = 28. Other perfect numbers are 496 and 8128. And they’re perfect in any base.

Amicable numbers are amicable in any base too. The factors of an amicable number sum to a second number whose factors sum to the first number. So s(220) = 284, s(284) = 220. That pair may have been known to Pythagoras (c.570-c.495 BC), but s(1184) = 1210, s(1210) = 1184 was discovered by an Italian schoolboy called Nicolò Paganini in 1866. There are also sociable chains, in which s(n), s(s(n)), s(s(s(n))) create a chain of numbers that leads back to n, like this:

12496 → 14288 → 15472 → 14536 → 14264 → 12496 (c=5)

Or this:

14316 → 19116 → 31704 → 47616 → 83328 → 177792 → 295488 → 629072 → 589786 → 294896 → 358336 → 418904 → 366556 → 274924 → 275444 → 243760 → 376736 → 381028 → 285778 → 152990 → 122410 → 97946 → 48976 → 45946 → 22976 → 22744 → 19916 → 17716 → 14316 (c=28)

Those sociable chains were discovered (and christened) in 1918 by the Belgian mathematician Paul Poulet (1887-1946). Other factor-sum patterns are dependant on the base they’re expressed in. For example, s(333) = 161. So both n and s(n) are palindromes in base-10. Here are more examples — the numbers in brackets are the prime factors of n and s(n):

333 (3^2, 37) → 161 (7, 23)
646 (2, 17, 19) → 434 (2, 7, 31)
656 (2^4, 41) → 646 (2, 17, 19)
979 (11, 89) → 101 (prime)
1001 (7, 11, 13) → 343 (7^3)
3553 (11, 17, 19) → 767 (13, 59)
10801 (7, 1543) → 1551 (3, 11, 47)
11111 (41, 271) → 313 (prime)
18581 (17, 1093) → 1111 (11, 101)
31713 (3, 11, 31^2) → 15951 (3, 13, 409)
34943 (83, 421) → 505 (5, 101)
48484 (2^2, 17, 23, 31) → 48284 (2^2, 12071)
57375 (3^3, 5^3, 17) → 54945 (3^3, 5, 11, 37)
95259 (3, 113, 281) → 33333 (3, 41, 271)
99099 (3^2, 7, 11^2, 13) → 94549 (7, 13, 1039)
158851 (7, 11, 2063) → 39293 (prime)
262262 (2, 7, 11, 13, 131) → 269962 (2, 7, 11, 1753)
569965 (5, 11, 43, 241) → 196691 (11, 17881)
1173711 (3, 7, 11, 5081) → 777777 (3, 7^2, 11, 13, 37)

Note how s(656) = 646 and s(646) = 434. There’s an even longer sequence in base-495:

33 → 55 → 77 → 99 → [17][17] → [19][19] → [21][21] → [43][43] → [45][45] → [111][111] → [193][193] → [195][195] → [477][477] (b=495) (c=13)
1488 (2^4, 3, 31) → 2480 (2^4, 5, 31) → 3472 (2^4, 7, 31) → 4464 (2^4, 3^2, 31) → 8432 (2^4, 17, 31) → 9424 (2^4, 19, 31) → 10416 (2^4, 3, 7, 31) → 21328 (2^4, 31, 43) → 22320 (2^4, 3^2, 5, 31) → 55056 (2^4, 3, 31, 37) → 95728 (2^4, 31, 193) → 96720 (2^4, 3, 5, 13, 31) → 236592 (2^4, 3^2, 31, 53)

I also tried looking for n whose s(n) mirrors n. But they’re hard to find in base-10. The first example is this:

498906 (2, 3^3, 9239) → 609894 (2, 3^2, 31, 1093)

498906 mirrors 609894, because the digits of each run in reverse to the digits of the other. Base-9 does better for mirror-sums, clocking up four in the same range of integers:

42 → 24 (base=9)
38 (2, 19) → 22 (2, 11)
402 → 204 (base=9)
326 (2, 163) → 166 (2, 83)
4002 → 2004 (base=9)
2918 (2, 1459) → 1462 (2, 17, 43)
5544 → 4455 (base=9)
4090 (2, 5, 409) → 3290 (2, 5, 7, 47)

Base-11 does better still, clocking up eight in the same range:

42 → 24 (base=11)
46 (2, 23) → 26 (2, 13)
2927 → 7292 (base=11)
3780 (2^2, 3^3, 5, 7) → 9660 (2^2, 3, 5, 7, 23)
4002 → 2004 (base=11)
5326 (2, 2663) → 2666 (2, 31, 43)
13772 → 27731 (base=11)
19560 (2^3, 3, 5, 163) → 39480 (2^3, 3, 5, 7, 47)
4[10]7[10]9 → 9[10]7[10]4 (base=11)
72840 (2^3, 3, 5, 607) → 146040 (2^3, 3, 5, 1217)
6929[10] → [10]9296 (base=11)
100176 (2^4, 3, 2087) → 158736 (2^4, 3, 3307)
171623 → 326171 (base=11)
265620 (2^2, 3, 5, 19, 233) → 520620 (2^2, 3, 5, 8677)
263702 → 207362 (base=11)
414790 (2, 5, 41479) → 331850 (2, 5^2, 6637)

Note that 42 mirrors its factor-sum in both base-9 and base-11. But s(42) = 24 in infinitely many bases, because when 42 = 2 x prime, s(42) = 1 + 2 + prime. So (prime-1) / 2 will give the base in which 24 = s(42). For example, 2 x 11 = 22 and 22 = 42 in base (11-1) / 2 or base-5. So s(42) = 1 + 2 + 11 = 14 = 2 x 5 + 4 = 24[b=5]. There are infinitely many primes, so infinitely many bases in which s(42) = 24.

Base-10 does better for mirror-sums when s(n) is re-defined to include n itself. So s(69) = 1 + 3 + 23 + 69 = 96. Here are the first examples of all-factor mirror-sums in base-10:

69 (3, 23) → 96 (2^5, 3)
276 (2^2, 3, 23) → 672 (2^5, 3, 7)
639 (3^2, 71) → 936 (2^3, 3^2, 13)
2556 (2^2, 3^2, 71) → 6552 (2^3, 3^2, 7, 13)

In the same range, base-9 now produces one mirror-sum, 13 → 31 = 12 (2^2, 3) → 28 (2^2, 7). Base-11 produces no mirror-sums in the same range. Base behaviour is eccentric, but that’s what makes it interesting.

Watch this Sbase

by in Mathematics and tagged , , , , , , , , , , , | Leave a comment

In standard notation, there are two ways to represent 2: 10, in base 2, and 2 in every other base. Accordingly, there are three ways to represent 3: 11 in base 2, 10 in base 3, and 3 in every other base. There are four ways to represent 4, five ways to represent 5, and so on. Now, suppose you sum all the digits of all the representations of n in the bases 2 to n, like this:

Σ(2) = 1+02 = 1
Σ(3) = 1+12 + 1+03 = 3 (+2)
Σ(4) = 1+0+02 + 1+13 + 1+04 = 4 (+1)
Σ(5) = 1+0+12 + 1+23 + 1+14 + 1+05 = 8 (+4)
Σ(6) = 1+1+02 + 2+03 + 1+24 + 1+15 + 1+06 = 10 (+2)
Σ(7) = 1+1+12 + 2+13 + 1+34 + 1+25 + 1+16 + 1+07 = 16 (+6)
Σ(8) = 1+0+0+02 + 2+23 + 2+04 + 1+35 + 1+26 + 1+17 + 1+08 = 17 (+1)
Σ(9) = 1+0+0+12 + 1+0+03 + 2+14 + 1+45 + 1+36 + 1+27 + 1+18 + 1+09 = 21 (+4)
Σ(10) = 1+0+1+02 + 1+0+13 + 2+24 + 2+05 + 1+46 + 1+37 + 1+28 + 1+19 + 1+010 = 25 (+4)

It seems reasonable to suppose that as n increases, so the all-digit-sum of n increases. But that isn’t always the case: occasionally it decreases. Here are the sums for n=11..100 (with prime factors when the sum is composite):

Σ(11) = 35 = 5·7 (+10)
Σ(12) = 34 = 2·17 (-1)
Σ(13) = 46 = 2·23 (+12)
Σ(14) = 52 = 22·13 (+6)
Σ(15) = 60 = 22·3·5 (+8)
Σ(16) = 58 = 2·29 (-2)
Σ(17) = 74 = 2·37 (+16)
Σ(18) = 73 (-1)
Σ(19) = 91 = 7·13 (+18)
Σ(20) = 92 = 22·23 (+1)
Σ(21) = 104 = 23·13 (+12)
Σ(22) = 114 = 2·3·19 (+10)
Σ(23) = 136 = 23·17 (+22)
Σ(24) = 128 = 27 (-8)
Σ(25) = 144 = 24·32 (+16)
Σ(26) = 156 = 22·3·13 (+12)
Σ(27) = 168 = 23·3·7 (+12)
Σ(28) = 171 = 32·19 (+3)
Σ(29) = 199 (+28)
Σ(30) = 193 (-6)
Σ(31) = 223 (+30)
Σ(32) = 221 = 13·17 (-2)
Σ(33) = 241 (+20)
Σ(34) = 257 (+16)
Σ(35) = 281 (+24)
Σ(36) = 261 = 32·29 (-20)
Σ(37) = 297 = 33·11 (+36)
Σ(38) = 315 = 32·5·7 (+18)
Σ(39) = 339 = 3·113 (+24)
Σ(40) = 333 = 32·37 (-6)
Σ(41) = 373 (+40)
Σ(42) = 367 (-6)
Σ(43) = 409 (+42)
Σ(44) = 416 = 25·13 (+7)
Σ(45) = 430 = 2·5·43 (+14)
Σ(46) = 452 = 22·113 (+22)
Σ(47) = 498 = 2·3·83 (+46)
Σ(48) = 472 = 23·59 (-26)
Σ(49) = 508 = 22·127 (+36)
Σ(50) = 515 = 5·103 (+7)
Σ(51) = 547 (+32)
Σ(52) = 556 = 22·139 (+9)
Σ(53) = 608 = 25·19 (+52)
Σ(54) = 598 = 2·13·23 (-10)
Σ(55) = 638 = 2·11·29 (+40)
Σ(56) = 634 = 2·317 (-4)
Σ(57) = 670 = 2·5·67 (+36)
Σ(58) = 698 = 2·349 (+28)
Σ(59) = 756 = 22·33·7 (+58)
Σ(60) = 717 = 3·239 (-39)
Σ(61) = 777 = 3·7·37 (+60)
Σ(62) = 807 = 3·269 (+30)
Σ(63) = 831 = 3·277 (+24)
Σ(64) = 819 = 32·7·13 (-12)
Σ(65) = 867 = 3·172 (+48)
Σ(66) = 861 = 3·7·41 (-6)
Σ(67) = 927 = 32·103 (+66)
Σ(68) = 940 = 22·5·47 (+13)
Σ(69) = 984 = 23·3·41 (+44)
Σ(70) = 986 = 2·17·29 (+2)
Σ(71) = 1056 = 25·3·11 (+70)
Σ(72) = 1006 = 2·503 (-50)
Σ(73) = 1078 = 2·72·11 (+72)
Σ(74) = 1114 = 2·557 (+36)
Σ(75) = 1140 = 22·3·5·19 (+26)
Σ(76) = 1155 = 3·5·7·11 (+15)
Σ(77) = 1215 = 35·5 (+60)
Σ(78) = 1209 = 3·13·31 (-6)
Σ(79) = 1287 = 32·11·13 (+78)
Σ(80) = 1263 = 3·421 (-24)
Σ(81) = 1293 = 3·431 (+30)
Σ(82) = 1333 = 31·43 (+40)
Σ(83) = 1415 = 5·283 (+82)
Σ(84) = 1368 = 23·32·19 (-47)
Σ(85) = 1432 = 23·179 (+64)
Σ(86) = 1474 = 2·11·67 (+42)
Σ(87) = 1530 = 2·32·5·17 (+56)
Σ(88) = 1530 = 2·32·5·17 (=)
Σ(89) = 1618 = 2·809 (+88)
Σ(90) = 1572 = 22·3·131 (-46)
Σ(91) = 1644 = 22·3·137 (+72)
Σ(92) = 1663 (+19)
Σ(93) = 1723 (+60)
Σ(94) = 1769 = 29·61 (+46)
Σ(95) = 1841 = 7·263 (+72)
Σ(96) = 1784 = 23·223 (-57)
Σ(97) = 1880 = 23·5·47 (+96)
Σ(98) = 1903 = 11·173 (+23)
Σ(99) = 1947 = 3·11·59 (+44)
Σ(100) = 1923 = 3·641 (-24)

The sum usually increases, occasionally decreases. In one case, when 87 = n = 88, it stays the same. This also happens when 463 = n = 464, where Σ(463) = Σ(464) = 39,375. Does it happen again? I don’t know. The ratio of sum-ups to sum-downs seems to tend towards 3:1. Is that the exact ratio at infinity? I don’t know. Watch this sbase.