small integers that are or might be the largest of their kind

small integers that are or might be the largest of their kind

1

is the largest

sum-product number

in binary, and the largest to be a sum-product number in any standard positional base. 1 is the largest

integer

whose

Zeckendorf representation

has no significant zeroes. Also, it is the largest (and the only) integer to be labelled non-prime without ever being a

probable prime

with unknown factorization.

2

is the largest (and the only)

even prime

number. Also, it might be the largest

n

such that no

n

×

n

magic square

consisting of consecutive primes can be constructed. The total of such magic squares is only known up to

n

=

6

.

3

is the largest and only prime

perfect totient number

.

4

is the largest

composite number

such that the union of the set of its

totatives

and the set of its

divisors

is the

complete

range of integers from 1 to itself. All larger numbers satisfying this

http://planetmath.org/RelationshipBetweenTotativesAndDivisors

relationship between their totatives and divisors are prime. In another commonality with primes, 4 might be the largest composite

n

such that

ϕ

⁢

(

n

)

⁢

σ

0

⁢

(

n

)

+

2

is a

multiple

of

n

, with

ϕ

⁢

(

n

)

being Euler’s

totient function

and

σ

0

⁢

(

n

)

being a count of the divisors. (For a prime

p

,

ϕ

⁢

(

p

)

⁢

σ

0

⁢

(

p

)

+

2

=

2

⁢

p

).

5

might be the largest

untouchable number

to be odd and prime. If a larger, odd though composite untouchable number were to be found, 5 would retain the distinction of being the largest such prime. Finding a larger prime untouchable number would strip 5 of both distinctions.

6

is the largest integer to be both a

factorial

and a

primorial

, and it’s the largest

squarefree

factorial. 6 is the largest

n

for which the inequality

ϕ

⁢

(

n

)

>

n

is false. Also, it is a

Harshad number

regardless of the base in any standard positional base notation. 6 is 110 in binary, 20 in ternary, 12 in base 4, 11 in base 5 and 10 in its own base, and then 6 in all bases afterwards. It is not palindromic in binary, ternary or quaternary, and is thus the largest composite

strictly non-palindromic number

.

8

is the largest

Fibonacci number

that is also a cube.

9

is the largest composite center of a

prime quadruplet

that is not a multiple of 15.

19

is the largest prime Roman numeral

palindromic number

(XIX). If one allows overlines, one would have to ignore the

symmetry

of the overlines in order to permit for a larger prime

palindrome

.

23

is the largest integer to have the same

representation

in both

factorial base

and primorial base (specifically, 321).

24

is the largest

n

such that

m

|

n

for all

0

<

m

<

n

. Also, it is the largest integer to satisfy the equality

∑

i

=

1

n

i

2

=

m

2

,

where

m

is an integer. [Tattersall, 2005]

26

is the largest (and only) integer sandwiched between a square and a cube.

30

is the largest integer such that none of its totatives are composite,

http://planetmath.org/EveryPositiveIntegerGreaterThan30HasAtLeastOneCompositeTotative

all greater integers have at least one composite totative. The count of those totatives happens to be equal to the count of its divisors, 30 is the largest integer for which this is true.

41

is the largest

n

such that the

polynomial

m

2

-

m

+

n

yields primes for any

positive

m

<

n

.

46

is the largest

even integer

for which there is no pair of abudant numbers that add up to it. (See the empirical proof that every sufficiently large even integer can be expressed as the sum of a pair of

abundant numbers

).

55

is the largest Fibonacci number that is also a

triangular number

.

60

is thought to be the largest integer that does not admit to a representation under Chen’s

theorem

(as a sum of two distinct primes or a sum of a prime and a

semiprime

; see A100952 in Sloane’s OEIS).60 is also the largest

n

such that

π

⁢

(

n

)

<

ϕ

⁢

(

n

)

is false (with

π

⁢

(

x

)

being the

prime counting function

).

61

might be the largest

prime number

p

x

(where

x

, the index of

p

in an ordered list of the primes in ascending order,

x

=

π

⁢

(

x

)

) such that

p

x

|

p

x

+

1

p

x

+

2

+

1

.

71

is the largest

supersingular prime

.

90

is the largest

n

such that

ϕ

⁢

(

n

)

=

π

⁢

(

n

)

, where

ϕ

⁢

(

x

)

is Euler’s totient function and

π

⁢

(

x

)

is the prime-counting function.

127

might be the largest prime

p

satisfying the three conditions of the new Mersenne conjecture [Ribenboim, 2004]. Also, it might be the largest prime

p

such that

2

p

-

1

is a

Chen prime

.

144

is the largest Fibonacci number that is also a square. In base 10 it is the largest sum-product number, a fact that is amazing when you consider that in order to prove it so David Wilson had to test sum-product number

candidates

as large as

10

84

.

163

is the largest Heegner

discriminant

.

216

might be the largest integer which is not the sum of a prime number and a triangular number. (Sun, 2008)

454

is the largest integer such that its shortest

partition

into cubes requires eight of them.

563

might be the largest

Wilson prime

.

786

might be the largest integer for which

C

n

2

⁢

n

is not

divisible

by the square of an

odd prime

.

1493

might be the largest

Stern prime

, since there is no way to put it in the form

p

+

2

⁢

b

2

, where

p

is a different prime and

b

>

0

. The first

hundred

thousand primes have been checked and all greater than 1493 can be put into the given form.

1806

is the largest

n

such that

m

n

+

1

=

m

mod

n

for any

m

.

For the purpose of this feature, the arbitrary cutoff is

10

4

.

References

1

R. K. Guy,

Unsolved Problems in

Number Theory

, B37. New York: Springer-Verlag (2004)

2

P. Ribenboim,

The Little Book of Bigger Primes

, p. 83. New York: Springer-Verlag (2004)

3

J. J. Tattersall,

Elementary number theory

in nine chapters

, p. 58. Cambridge: Cambridge University Press (2005)

4

Zhi-Wei Sun, “On Sums of Primes and Triangular Numbers” ArXiv preprint (2008): 2

Title

small integers that are or might be the largest of their kind

Canonical

name

SmallIntegersThatAreOrMightBeTheLargestOfTheirKind

Date of creation

2013-03-22 15:53:04

Last modified on

2013-03-22 15:53:04

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

22

Author

Mravinci (12996)

Entry type

Feature

Classification

msc 00A08

Related topic

LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind

Related topic

EveryPositiveIntegerGreaterThan30HasAtLeastOneCompositeTotative

Generated on Tue Feb 6 22:19:38 2018 by

LaTeXML