Untouchable quantity – Wikipedia

Quantity that can not be written as an aliquot sum

Unsolved downside in arithmetic:

Are there any odd untouchable numbers aside from 5?

An untouchable quantity is a constructive integer that can not be expressed because the sum of all of the proper divisors of any constructive integer (together with the untouchable quantity itself). That’s, these numbers aren’t within the picture of the aliquot sum operate. Their examine goes again at the least to Abu Mansur al-Baghdadi (circa 1000 AD), who noticed that each 2 and 5 are untouchable.^[1]

Examples[edit]

The quantity 4 shouldn’t be untouchable because it is the same as the sum of the right divisors of 9: 1 + 3 = 4. The quantity 5 is untouchable as it isn’t the sum of the right divisors of any constructive integer: 5 = 1 + 4 is the one approach to write 5 because the sum of distinct constructive integers together with 1, but when 4 divides a quantity, 2 does additionally, so 1 + 4 can’t be the sum of all of any quantity’s correct divisors (because the listing of things must comprise each 4 and a couple of).

The primary few untouchable numbers are

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, … (sequence A005114 within the OEIS).

Properties[edit]

The quantity 5 is believed to be the one odd untouchable quantity, however this has not been confirmed. It might observe from a barely stronger model of the Goldbach conjecture, because the sum of the right divisors of pq (with p, q distinct primes) is 1 + p + q. Thus, if a quantity n may be written as a sum of two distinct primes, then n + 1 shouldn’t be an untouchable quantity. It’s anticipated that each even quantity bigger than 6 is a sum of two distinct primes, so most likely no odd quantity bigger than 7 is an untouchable quantity, and ${displaystyle 1=sigma (2)-2}$ , ${displaystyle 3=sigma (4)-4}$ , ${displaystyle 7=sigma (8)-8}$ , so solely 5 may be an odd untouchable quantity.^[2] Thus it seems that in addition to 2 and 5, all untouchable numbers are composite numbers (since besides 2, all even numbers are composite). No perfect number is untouchable, since, on the very least, it may be expressed because the sum of its personal correct divisors. Equally, not one of the amicable numbers or sociable numbers are untouchable. Additionally, not one of the Mersenne numbers are untouchable, since M_n = 2ⁿ − 1 is the same as the sum of the right divisors of twoⁿ.

No untouchable quantity is another than a prime number, since if p is prime, then the sum of the right divisors of p² is p + 1. Additionally, no untouchable quantity is three greater than a main quantity, besides 5, since if p is an odd prime then the sum of the right divisors of twop is p + 3.

Infinitude[edit]

There are infinitely many untouchable numbers, a indisputable fact that was confirmed by Paul Erdős.^[3] Based on Chen & Zhao, their natural density is at the least d > 0.06.^[4]

See additionally[edit]

References[edit]

^ Sesiano, J. (1991), “Two issues of quantity idea in Islamic instances”, Archive for Historical past of Actual Sciences, 41 (3): 235–238, doi:10.1007/BF00348408, JSTOR 41133889, MR 1107382, S2CID 115235810
^ The stronger model is obtained by including to the Goldbach conjecture the additional requirement that the 2 primes be distinct—see Adams-Watters, Frank & Weisstein, Eric W. “Untouchable Number”. MathWorld.
^ P. Erdos, Über die Zahlen der Type ${displaystyle sigma (n)-n}$ und ${displaystyle n-phi (n)}$ . Elemente der Math. 28 (1973), 83-86
^ Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen 78:2 (2011), pp. 439-442.