In mathematics, a noncototient is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...

integers below it. That is, ''m'' − φ(''m'') = ''n'', where φ stands for

Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...

, has no solution for ''m''. The '' cototient'' of ''n'' is defined as ''n'' − φ(''n''), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number ''n'' can be represented as a sum of two distinct primes ''p'' and ''q,'' then :

pq - \varphi(pq) = pq - (p-1)(q-1) = p+q-1 = n-1. \,

It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations

1=2-\phi(2), 3 = 9 - \phi(9)

and

5 = 25 - \phi(25)

. For even numbers, it can be shown :

2pq - \varphi(2pq) = 2pq - (p-1)(q-1) = pq+p+q-1 = (p+1)(q+1)-2

Thus, all even numbers ''n'' such that ''n''+2 can be written as (p+1)*(q+1) with ''p'', ''q'' primes are cototients. The first few noncototients are : 10, 26, 34, 50, 52, 58, 86, 100, 116,

122 122 may refer to: *122 (number), a natural number * AD 122, a year in the 2nd century AD * 122 BC, a year in the 2nd century BC * ''122'' (film), a 2019 Egyptian psychological horror film *"One Twenty Two", a 2022 single by the American rock band Bo ...

, 130, 134, 146,

154 Year 154 ( CLIV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Lateranus (or, less frequently, year 907 ''Ab urbe cond ...

, 170, 172, 186, 202, 206, 218,

222 __NOTOC__ Year 222 (Roman numerals, CCXXII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Antoninus and Severus (or, less ...

, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, ... The cototient of ''n'' are :0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 46, 1, 48, ... Least ''k'' such that the cototient of ''k'' is ''n'' are (start with ''n'' = 0, 0 if no such ''k'' exists) :1, 2, 4, 9, 6, 25, 10, 15, 12, 21, 0, 35, 18, 33, 26, 39, 24, 65, 34, 51, 38, 45, 30, 95, 36, 69, 0, 63, 52, 161, 42, 87, 48, 93, 0, 75, 54, 217, 74, 99, 76, 185, 82, 123, 60, 117, 66, 215, 72, 141, 0, ... Greatest ''k'' such that the cototient of ''k'' is ''n'' are (start with ''n'' = 0, 0 if no such ''k'' exists) :1, ∞, 4, 9, 8, 25, 10, 49, 16, 27, 0, 121, 22, 169, 26, 55, 32, 289, 34, 361, 38, 85, 30, 529, 46, 133, 0, 187, 52, 841, 58, 961, 64, 253, 0, 323, 68, 1369, 74, 391, 76, 1681, 82, 1849, 86, 493, 70, 2209, 94, 589, 0, ... Number of ''k''s such that ''k''-φ(''k'') is ''n'' are (start with ''n'' = 0) :1, ∞, 1, 1, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 2, 3, 3, 1, 3, 1, 3, 1, 4, 4, 3, 0, 4, 1, 4, 3, 3, 4, 3, 0, 5, 2, 2, 1, 4, 1, 5, 1, 4, 2, 4, 2, 6, 5, 5, 0, 3, 0, 6, 2, 4, 2, 5, 0, 7, 4, 3, 1, 8, 4, 6, 1, 3, 1, 5, 2, 7, 3, ... Erdős (1913-1996) and Sierpinski (1882-1969) asked whether there exist infinitely many noncototients. This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family

2^k \cdot 509203

is an example (See

Riesel number In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the f ...

). Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca (2000).

References

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External links

Noncototient definition from MathWorld
{{Classes of natural numbers Integer sequences