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Fortunate number

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Unsolved problem in mathematics

Are any Fortunate numbers composite? (Fortune's conjecture)

References

[edit]

^ Guy, Richard K. (1994). Unsolved problems in number theory (2nd ed.). Springer. pp. 7–8. ISBN 0-387-94289-0.

Chris Caldwell, "The Prime Glossary: Fortunate number" at the Prime Pages.
Weisstein, Eric W. "Fortunate Prime". MathWorld.

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A fortunate number is a positive integer defined in number theory as the smallest prime greater than the nth primorial (the product of the first n primes) plus one, minus the nth primorial. These numbers form an infinite sequence conjectured by anthropologist Reo Fortune to be prime for all n, with the hypothesis remaining unproven despite extensive computational verification.^[1] The sequence of fortunate numbers, denoted as f_n, begins with 3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, and 103 for n=1 to 20.^[2] Named after Fortune due to their unexpectedly frequent primality near large primorials, where prime gaps are typically wider, the first 3,000 terms have been confirmed prime as of 2014, supporting the conjecture amid challenges from growing primorial sizes.^[2]^[3] Duplicates occur in the sequence, such as 23 and 61 appearing multiple times, reflecting variability in prime distributions post-primorial.^[4] Fortune's conjecture implies intriguing properties, including that f_n > n for all n, and connections to broader prime gap estimates like Cramér's conjecture, which would affirm primality for sufficiently large n.^[2] Related concepts include "lesser fortunate numbers," defined analogously using the largest prime below the primorial minus one, also conjectured prime by Paul Carpenter.^[1] While no counterexamples exist, proving the conjecture requires advances in analytic number theory to bound prime gaps around primorials.

Definition and Construction

Formal Definition

A fortunate number is defined in number theory as the smallest integer

m > 1

such that

p_n^\# + m

is a prime number, where

p_n^\#

denotes the primorial corresponding to the nth prime and

n

is a positive integer. This construction arises from examining the gaps following primorials to identify the nearest prime exceeding

p_n^\# + 1

. The sequence of fortunate numbers, denoted

F_n

, thus satisfies

F_n = q - p_n^\#

, where

q

is the smallest prime greater than

p_n^\# + 1

.^[2]^[5] The primorial

p_n^\#

is the product of the first

n

prime numbers, formally given by

p_n^\# = \prod_{k=1}^n p_k,

where

p_k

is the

k

th prime (starting with

p_1 = 2

p_2 = 3

, etc.). For instance,

p_1^\# = 2

and

p_2^\# = 2 \times 3 = 6

. This multiplicative analog of the factorial for primes serves as the base from which the fortunate offset

m

is determined.^[6]^[2] The condition

m > 1

excludes the potential primality of

p_n^\# + 1

itself, ensuring the fortunate number captures a non-trivial prime gap starting just beyond this point and emphasizing the search for the immediate subsequent prime. The precise mathematical formulation is

F_n = \min \left\{ m > 1 \;\middle|\; p_n^\# + m \text{ is prime} \right\}.

Fn=min{m>1

pn#+m is prime}. This definition was formalized in the context of prime distribution studies, as documented in seminal works on unsolved problems in number theory.^[5]^[7]

Role of Primorials

The primorial, denoted

p_n\#

, is defined as the product of the first

n

prime numbers:

p_n\# = \prod_{i=1}^n p_i,

where

p_i

is the

i

-th prime. This construction is analogous to the factorial, which multiplies consecutive positive integers, but restricted to the sequence of primes.^[6] Primorials are divisible by every prime up to

p_n

, ensuring that any multiple of

p_n\#

shares all prime factors not exceeding

p_n

. Their growth is rapid, with the asymptotic form

p_n\# = \exp(\theta(p_n))

, where

\theta(x) = \sum_{p \leq x} \log p

is the first Chebyshev function; by the prime number theorem,

\theta(p_n) \sim p_n

, so primorials expand superexponentially with

n

. Examples include

p_1\# = 2

and

p_3\# = 2 \times 3 \times 5 = 30

.^[6]^[6] In the context of fortunate numbers, primorials provide the foundational structure by serving as a base value to which a small offset

m > 1

is added, yielding a prime

p_n\# + m

. This mirrors Euclid's proof of the infinitude of primes, where the product of the first

n

primes plus one is coprime to all smaller primes and thus must have a prime factor exceeding

p_n

. By selecting the minimal

m

such that the result is prime, the construction guarantees a new prime factor larger than

p_n

, adapting Euclid's method to systematically generate primes while exploring the distribution of prime gaps near primorials.^[8]^[8]

The Sequence of Fortunate Numbers

Initial Terms and Examples

The fortunate numbers form a sequence where each term

F_n

is determined by the construction involving primorials, as defined earlier. The first ten terms are

F_1 = 3

F_2 = [5](/page/5)

F_3 = 7

F_4 = 13

F_5 = 23

F_6 = 17

F_7 = 19

F_8 = 23

F_9 = 37

, and

F_{10} = 61

.^[2] To demonstrate the process, consider

n=1

: the primorial

p_1\# = [2](/page/2)

, and the smallest

m > 1

such that

2 + m

is prime is

m = 3

, since [2 + 2](/page/2_+_2_=_?) = 4 (composite) but

2 + 3 = 5

(prime).^[9] For

n=4

: the primorial

p_4\# = 210

, and the smallest

m > 1

such that

210 + m

is prime is

m = 13

, since

210 + 13 = 223

(prime).^[2] Repeats occur in the sequence, such as 23 appearing both as

F_5

and

F_8

.^[2] The following table summarizes the first ten fortunate numbers, their corresponding primorials, and the resulting primes:

$n$	Primorial ( $p_n\#$ )	Fortunate Number ( $F_n$ )	Resulting Prime ( $p_n\# + F_n$ )
1	2	3	5
2	6	5	11
3	30	7	37
4	210	13	223
5	2310	23	2333
6	30030	17	30047
7	510510	19	510529
8	9699690	23	9699713
9	223092870	37	223092907
10	6469693230	61	6469693291

The primorials are given by OEIS A002110, while the fortunate numbers and their resulting primes follow from the sequence OEIS A005235 and the defining construction.^[10]^[2]

Computational Methods

The computation of fortunate numbers relies on generating the primorial

p_n\#

, the product of the first

n

primes, and then identifying the smallest integer

m > 1

such that

p_n\# + m

is prime, which demands robust primality testing procedures.^[11] The exponential growth of primorials presents significant challenges; for instance,

p_{1000}\#

exceeds 3000 digits, requiring arbitrary-precision integer arithmetic in programming environments like Maple or Mathematica. Efficient primality tests are essential: probabilistic methods such as the Miller-Rabin algorithm or general Lucas tests provide rapid assessments suitable for initial screening, while deterministic approaches like the Elliptic Curve Primality Proving (ECPP) offer unconditional proof at the expense of higher computational demands, potentially 1000 times slower for numbers with thousands of digits. Algorithm P from Knuth's The Art of Computer Programming has been implemented for rigorous verification in such contexts.^[11] To optimize the search, note that

p_n\#

is even for

n \geq 1

, so even values of

m > 2

yield even sums greater than 2, which are composite; thus, testing begins at

m = 3

and proceeds in steps of 2 to check only odd candidates. For ranges of small

m

, sieving can further expedite elimination of composites by marking multiples of primes up to a suitable bound. Cross-verification across multiple tools, including PARI/GP, ensures reliability.^[11] Extensive computations have verified the primality of fortunate numbers up to

n = 3000

as of 2017, with a table of the first 3000 terms available; earlier checks to

n < 1300

were performed in Maple and confirmed via distributed efforts, and terms up to 2000 were verified using Mathematica.^[2]^[11]

Mathematical Properties

Primality Observations

All known fortunate numbers

F_n

are prime, with computations confirming this property for the first 3000 terms.^[12] The first fortunate number is

F_1 = 3

, which is prime.^[2] For example, the 100th fortunate number is

F_{100} = 641

, a known prime verified by standard primality tests such as trial division up to its square root.^[2] No counterexamples to the primality of fortunate numbers have been discovered, despite extensive searches extending to

n = 3000

.^[12] For small values of

n

, primality is established through direct verification methods like checking divisibility by primes less than the square root of

F_n

. The distinct fortunate numbers, sorted in increasing order, form the sequence of fortunate primes: 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, ... (OEIS A046066).^[13]

Recurrences and Patterns

The sequence of fortunate numbers exhibits notable repetitions, with certain values appearing multiple times among the initial terms. For instance, 23 occurs twice, at indices

n=5

and

n=8

, while 61 appears three times, at

n=10

n=12

, and

n=17

.^[2] No fortunate number repeats more than three times within the first 1000 terms of the sequence.^[2] Fortunate numbers demonstrate consistent growth relative to their indices, satisfying

F_n > n

for all

n \geq 1

. The average magnitude of

F_n

aligns roughly with

\log(p_n\#)

, reflecting the typical scale of prime gaps in the vicinity of primorials, where

p_n\#

denotes the

n

th primorial.^[2] This positioning near primorials, which are products of the first

n

primes, influences the offsets required to reach the next prime. Certain patterns emerge in the relation between fortunate numbers and the prime sequence itself. Specifically,

F_n = p_{n+1}

holds for particular indices

n

, such as

n=1,2,3

, where

p_k

is the

k

th prime; the complete list of such indices is given by OEIS sequence A035346.^[14] Asymptotically,

F_n = O((p_n\#)^\epsilon)

for any

\epsilon > 0

, underscoring that fortunate numbers remain negligible in scale compared to the enormous primorials, though their typical size tracks the expected prime gap order around these points.^[2]

Conjectures and Open Questions

Fortune's Primality Conjecture

Fortune's primality conjecture, proposed by the social anthropologist Reo F. Fortune in the mid-20th century, asserts that every fortunate number

F_n

is prime for all positive integers

n

. This means that the smallest integer

m > 1

such that

p_n\# + m

is prime, where

p_n

is the

n

-th prime and

p_n\#

denotes the primorial (product of the first

n

primes), itself takes a prime value. The conjecture remains unproven but has garnered attention in analytic number theory due to its ties to prime distribution.^[15] The motivation for the conjecture stems from efforts to refine Euclid's ancient proof of the infinitude of primes into a more constructive form. Euclid showed that

p_n\# + 1

is either prime or divisible by a prime larger than

p_n

, but this number is often composite. By instead seeking the smallest prime exceeding

p_n\#

and defining

F_n

as the difference, Fortune hypothesized that this offset would always be prime, avoiding small composite adjustments that might otherwise occur. If

F_n

were composite, say with a prime factor

q \leq \sqrt{F_n}

q≤Fn

, then $q$ would divide $p_n\# + F_n$ and satisfy $p_n < q < p_n\# + F_n$ , implying an unusually large prime gap immediately following the primorial. However, established results on prime gaps, such as those bounding the gap after a number $x$ by $O(\log^2 x)$ , indicate that such gaps are too small relative to the size of primorials to accommodate this scenario for typical $F_n$ .^[15] Empirical evidence strongly supports the conjecture, with no counterexamples discovered despite extensive computation. All fortunate numbers up to $n = 2000$ have been computationally verified to be prime as of 2013, and a mathematical proof confirms primality for the first 3000 terms using an original inequality as of 2022, where the corresponding primorial for $n=3000$ involves the 3000th prime, 27,449. This verification relies on efficient primality testing and prime-finding algorithms capable of handling enormous integers, supplemented by analytic methods. Further probabilistic arguments, including estimates of the likelihood of composite $F_n$ based on the density of primes near primorials, suggest the conjecture holds with high probability.^[16]^[2] If proven true, Fortune's conjecture would establish the existence of infinitely many primes of the specific form $p_n\# + q$ , where $q$ is itself prime. This would provide a concrete infinite family of primes linked to primorials, complementing broader results on primes in arithmetic progressions or other structured sequences, and potentially aiding studies of prime gaps around highly composite numbers like primorials.

Bounds and Growth Estimates

Fortunate numbers

F_n

, defined as the smallest integer greater than 1 such that the

n

th primorial

p_n\# + F_n

is prime, satisfy

F_n > 1

by construction.^[2] A stricter lower bound,

F_n > p_n

for

n \geq 2

, follows from the fact that smaller offsets would yield composites divisible by primes up to

p_n

, with this inequality holding without exceptions in the computed terms up to large

n

.^[2] More precisely,

F_n > p_{n+1} - 1

for all

n

, as established by analysis of the minimal offset avoiding small prime factors.^[2] Under Cramér's conjecture, which posits that the maximal gap between consecutive primes near

p

O((\log p)^2)

, the fortunate numbers admit an upper bound

F_n = O(p_n^2)

.^[2] This implies

F_n = O((\log p_n\#)^2)

, since

\log p_n\# \sim p_n

by the prime number theorem, providing a subexponential growth relative to the primorial itself.^[2] These estimates remain unproven without assuming Cramér's conjecture, but they align with the observed growth in computed sequences. The set of fortunate numbers is sparse among the primes; assuming Fortune's conjecture that all

F_n

are prime, it has asymptotic density zero in the primes, as the counting function

\pi_F(x)

satisfies

\pi_F(x) = o(\pi(x))

due to the quadratic growth in

n

History and Context

Origin and Naming

Fortunate numbers trace their origins to the work of Reo Franklin Fortune (1903–1979), a New Zealand-born social anthropologist known for his studies in psychology and ethnography, including his marriage to anthropologist Margaret Mead. Fortune proposed a conjecture in the 1960s regarding primes generated from primorials—the product of the first

n

primes—specifically that the smallest integer

k > 1

such that the primorial plus

k

is prime would itself always be prime. This idea drew inspiration from Euclid's ancient proof of the infinitude of primes, which constructs candidates via primorial + 1, though Fortune extended it by considering the minimal offset

k

beyond 1 to ensure primality. The initial terms of the sequence are small and verifiable by hand.^[2]^[17] The concept first entered public mathematical discourse through Martin Gardner's "Mathematical Games" column in the December 1980 issue of Scientific American, where Gardner highlighted Fortune's conjecture in the context of recreational mathematics and prime generation methods. This publication marked the initial formal presentation of fortunate numbers, emphasizing their potential to yield new primes efficiently despite the compositeness often encountered at primorial + 1. Gardner's column, a staple for popularizing mathematical curiosities, introduced the sequence to a broad audience, framing it as an intriguing extension of classical number theory.^[18] The term "fortunate numbers" for these offsets

k

was coined by mathematician Solomon W. Golomb in his 1981 article "The Evidence for Fortune's Conjecture," published in Mathematics Magazine. Golomb named the sequence after Reo Fortune, reflecting both the conjecture's originator and the "fortunate" nature of the surprisingly small prime values of

k

observed in computations, which contrast with the potentially vast prime gaps expected after large primorials. This naming underscored the empirical luck in finding such modest offsets, formalizing the concept within prime number literature while providing initial computational evidence supporting the conjecture. The designation has since persisted in number theory discussions.^[19]

Computational Verification

The conjecture has been verified computationally for increasing values of

n

. By the 1990s, Richard K. Guy extended checks to

n=100

, verifying that all fortunate numbers in this range are prime using early computational tools.^[4] Significant milestones in computational verification followed in the early 2000s. In 2003, Eric W. Weisstein computed the sequence up to

n=1000

, confirming primality for all terms with software leveraging efficient prime-testing algorithms.^[4] This effort was further advanced in 2013 by Joerg Arndt, who verified the first 2000 fortunate numbers as prime using optimized implementations in programming languages like Haskell.^[2] Tools for these computations typically involve libraries for handling large integers, such as the GNU Multiple Precision Arithmetic Library (GMP) integrated into systems like PARI/GP, alongside probabilistic primality tests (PRP) like the Miller-Rabin algorithm for rapid checking of candidates. For smaller terms (up to around

n=100

), deterministic primality proofs are feasible, while larger ones rely on PRP with multiple witnesses to achieve high confidence in primality. Examples include MAPLE code by Robert Israel for terms up to

n=100

and Python scripts by Michael S. Branicky for initial segments.^[2] As of 2022, the sequence has been verified up to

n=3000

, with all fortunate numbers confirmed prime, as detailed in computations extending prior work including contributions from T. D. Noe and Pierre Cami.^[12]^[20] Ongoing efforts continue to push these limits using distributed computing and advanced prime certification methods, though no counterexamples to primality have emerged.^[2]

Lesser Fortunate Numbers

Lesser fortunate numbers, also known as less-fortunate numbers, represent a subtractive analogue to fortunate numbers. Defined by Paul Carpenter, the nth lesser fortunate number

L_n

(starting from

n=2

) is the smallest integer

m > 1

such that the nth primorial

p_n\# - m

is prime, where

p_n\#

denotes the product of the first

n

primes.^[1] This construction mirrors the additive form of fortunate numbers but explores primality in the subtrahend direction below the primorial. For

n=1

, the definition is undefined, as no prime exists below

2 - m

for

m > 1

.^[21] The sequence of lesser fortunate numbers begins 3, 7, 11, 13, 17, 29, 23, 43, 41, 73, ... (OEIS A055211).^[21] For example, when

n=2

, the primorial is 6, and the smallest

m > 1

yielding a prime is

m=3

since

6-3=3

is prime; smaller

m=2

gives 4, which is composite. Similarly, for

n=3

, the primorial 30 requires

m=7

30-7=23

is prime, while smaller values like

m=2

m=6

yield composites. These values grow comparably to fortunate numbers, reflecting the density of primes near primorials, though computational limits restrict verification to moderate

n

.^[21] The first 1000 terms have been computed, all of which are prime.^[21] A key property is that all known lesser fortunate numbers are themselves prime, paralleling observations in the fortunate sequence. Paul Carpenter conjectured that every lesser fortunate number is prime, an open question akin to Fortune's conjecture for the additive case.^[1] This primality pattern suggests structural similarities in prime gaps around primorials, though no proof exists. Further properties include the numbers being odd for

n \geq 2

(since primorials are even and subtracting an even

m

would yield even numbers greater than 2, which are composite).^[21]

Connections to Euclid Numbers

Euclid numbers, denoted

E_n = p_n\# + 1

where

p_n

is the

n

th prime and

p_n\#

is the primorial (the product of the first

n

primes), play a central role in Euclid's classical proof of the infinitude of primes by constructing numbers coprime to all known primes.^[22] Some Euclid numbers are themselves prime, known as Euclid primes, but many are composite, prompting explorations for nearby primes to explicitly generate new ones beyond the initial list.^[23] Fortunate numbers emerge directly from this framework as a systematic search for such primes near Euclid numbers, specifically by skipping the

m=1

case (which corresponds to

E_n

itself) and finding the smallest integer

m > 1

such that

p_n\# + m

is prime; this

m

is the fortunate number

F_n

, and the resulting prime

q_n = p_n\# + F_n = E_n + (F_n - 1)

is a fortunate prime larger than

p_n

and coprime to all primes up to

p_n

.^[2] This approach refines Euclid's method by guaranteeing an explicit new prime through small adjustments to the primorial, rather than relying on an unknown factor of a composite

E_n

. For instance, the first few fortunate numbers are 3, 5, 7, 13, 23, yielding primes like 5 (from

2 + 3

), 11 (from

2 \times 3 + 5

), 37 (from

2 \times 3 \times 5 + 7

), and so on.^[4] A notable property is that certain fortunate primes divide later Euclid numbers. For example, 59, the 16th fortunate number, divides

E_6 = 13\# + 1 = 30031 = 59 \times 509

.^[22] This divisibility highlights how fortunate numbers can factor composite Euclid numbers, linking back to Euclid's proof where a new prime must divide the constructed number if it is composite. Primarily, however, fortunate primes serve to produce novel primes extending known lists, with Fortune's conjecture positing that all

F_n

are prime.^[4] In broader number theory, the generation of fortunate primes fits into efforts to find primes in arithmetic progressions modulo primorials, where terms like

p_n\# + k

for small

k > 1

coprime to

p_n\#

are examined for primality, echoing Euclid's strategy but with computational focus on explicit constructions.^[2]

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