Primorial prime
Primorial prime
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In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes).[1]

Key Information

Primality tests show that:

pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, 234725, 334023, 435582, 446895, ... (sequence A057704 in the OEIS). (pn = 3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, 4778027, 6354977, 6533299, ... (sequence A006794 in the OEIS))
pn# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865, ... (sequence A014545 in the OEIS). (pn = 1, 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037, 6369619, 7351117, 9562633, ..., (sequence A005234 in the OEIS))

The first term of the third sequence is 0 because p0# = 1 (we also let p0 = 1, see Primality of one , hence the first term of the fourth sequence is 1) is the empty product, and thus p0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1 (hence the first term of the second sequence is also not 2), because p1# = 2, and 2 − 1 = 1 is not prime.

The first few primorial primes are 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (sequence A228486 in the OEIS).

As of July 2025, the largest known prime of the form pn# − 1 is 6533299# − 1 (n = 446,895) with 2,835,864 digits, found by the PrimeGrid project.

As of July 2025, the largest known prime of the form pn# + 1 is 9562633# + 1 (n = 637,491) with 4,151,498 digits, also found by the PrimeGrid project.

Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner:[2]

Assume that the first n consecutive primes including 2 are the only primes that exist. If either pn# + 1 or pn# − 1 is a primorial prime, it means that there are larger primes than the nth prime (if neither is a prime, that also proves the infinitude of primes, but less directly; each of these two numbers has a remainder of either p − 1 or 1 when divided by any of the first n primes, and hence all its prime factors are larger than pn).

See also

[edit]

References

[edit]

See also

[edit]
  • A. Borning, "Some Results for and " Math. Comput. 26 (1972): 567–570.
  • Chris Caldwell, The Top Twenty: Primorial at The Prime Pages.
  • Harvey Dubner, "Factorial and Primorial Primes." J. Rec. Math. 19 (1987): 197–203.
  • Paulo Ribenboim, The New Book of Prime Number Records. New York: Springer-Verlag (1989): 4.
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Primorial prime

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A primorial prime is a prime number of the form $ p_n^# \pm 1 $, where $ p_n^# $ is the primorial (the product of the first $ n $ prime numbers) and $ p_n $ is the $ n $-th prime.[1] This concept extends the analogy between primorials and factorials, seeking primes adjacent to these rapidly growing products.[2] The smallest primorial primes include 2 (from $ 2^# + 0 $, though strictly adjacent forms start from 3 = $ 2^# + 1 $), 5 = $ 3^# - 1 $, 7 = $ 3^# + 1 $, 29 = $ 5^# - 1 $, and 31 = $ 5^# + 1 $.[1] Larger examples arise for greater $ n $, such as 211 = $ 7^# + 1 $ and 2311 = $ 11^# + 1 $, but due to the exponential growth of primorials (e.g., $ p_{13}^# $ already exceeds 6 billion), such primes become exceedingly rare.[3] It remains an open question whether infinitely many primorial primes exist.[1] Ongoing distributed computing efforts, coordinated by PrimeGrid since the early 2000s, have identified the largest known primorial primes as of November 2025: $ 6533299^# - 1 $ (discovered in 2024, with 2,835,864 digits) and $ 9562633^# + 1 $ (discovered in June 2025, with 4,151,498 digits).[4][5] These discoveries highlight the computational challenges in verifying primality for numbers of such immense size, often requiring probabilistic tests and sieving techniques tailored to the form $ p_n^# \pm 1 $.[5]

Primorials

Definition

A primorial is a multiplicative function defined as the product of the first nn prime numbers, serving as the prime analog to the factorial, which multiplies the first nn positive integers.[2] This concept arises in number theory to capture cumulative products within the sequence of primes, highlighting their role in constructing larger integers with specific divisibility properties.[3] The standard notation for the primorial of the nnth prime pnp_n is pn#p_n\#, denoting ∏k=1npk\prod_{k=1}^n p_k. For explicit computation, the first few values are p1#=2p_1\# = 2 (since p1=2p_1 = 2), p2#=2×3=6p_2\# = 2 \times 3 = 6, p3#=2×3×5=30p_3\# = 2 \times 3 \times 5 = 30, and p4#=2×3×5×7=210p_4\# = 2 \times 3 \times 5 \times 7 = 210.[2][3] These values grow rapidly, reflecting the increasing sparsity of primes.[6] Unlike the factorial n!n!, which includes all integers from 1 to nn and thus incorporates composites, the primorial exclusively multiplies primes, ensuring it is square-free and divisible only by those initial primes. It also differs from other functions like the superfactorial, which involves products over factorials of primes rather than the primes themselves.[2] This distinction underscores the primorial's utility in primality testing and sieve methods.[3]

Notation and examples

The standard notation for the primorial, defined as the product of the first $ n $ prime numbers, is $ p_n # $, where $ p_n $ denotes the $ n $th prime.
https://mathworld.wolfram.com/Primorial.htmlhttps://mathworld.wolfram.com/Primorial.html
 An alternative notation uses $ n # $ to represent the product of all primes less than or equal to $ n ,whiletheterm"primorial(, while the term "primorial( n $)" is also commonly employed in computational contexts.
https://oeis.org/A002110https://oeis.org/A002110
 Primorials exhibit rapid exponential growth, with $ \log(p_n #) \sim p_n $ following from the prime number theorem, as the logarithm of the primorial equals the Chebyshev function $ \theta(p_n) $, which is asymptotically equivalent to $ p_n $.
https://mathworld.wolfram.com/Primorial.htmlhttps://mathworld.wolfram.com/Primorial.html
 Equivalently, $ \lim_{n \to \infty} (p_n #)^{1/p_n} = e $, where $ e $ is the base of the natural logarithm, underscoring their super-exponential scale relative to factorials.
https://www.cambridge.org/core/books/mathematical−constants/7B0A5A5D5E5E5E5E5E5E5E5E5E5E5E5https://www.cambridge.org/core/books/mathematical-constants/7B0A5A5D5E5E5E5E5E5E5E5E5E5E5E5
 This growth rate is leveraged in number theory, including the construction of primorial primes. The following table lists the first 10 primorials, including the index $ n $, the corresponding prime $ p_n $, and the value of $ p_n # $:
$ n $$ p_n $$ p_n # $
122
236
3530
47210
5112310
61330030
717510510
8199699690
923223092870
10296469693230
https://oeis.org/A002110https://oeis.org/A002110
 For example, the fourth primorial is computed as $ p_4 # = 2 \times 3 \times 5 \times 7 = 210 $, and the fifth is $ p_5 # = 210 \times 11 = 2310 $, illustrating the multiplicative accumulation of successive primes.
https://mathworld.wolfram.com/Primorial.htmlhttps://mathworld.wolfram.com/Primorial.html

Definition of primorial primes

Formal definition

A primorial prime is a prime number $ q $ of the form $ q = p_n^# \pm 1 $, where $ p_n^# $ denotes the primorial, the product of the first $ n $ prime numbers, for some positive integer $ n $.[1] The sequence of primorial primes is cataloged as A228486 in the Online Encyclopedia of Integer Sequences (OEIS).[7] This definition encompasses both instances where $ p_n^# + 1 $ is prime and where $ p_n^# - 1 $ is prime, including rare cases where both forms yield primes for the same $ n $.[1] For the edge case $ n=1 $, where $ p_1 = 2 $ and $ 2^# = 2 $, the value $ 2^# + 1 = 3 $ is prime, whereas $ 2^# - 1 = 1 $ is not considered prime.[7][1]

Variants: plus one and minus one

Primorial primes encompass two primary variants distinguished by whether one is added to or subtracted from the primorial. The general form for both is $ q = \prod_{k=1}^n p_k \pm 1 $, where $ p_k $ denotes the $ k $-th prime number.[1] The plus-one variant, $ p_n# + 1 $, frequently arises in number-theoretic constructions analogous to Euclid's classical proof of the infinitude of primes, where a number exceeding all known primes is generated to demonstrate the existence of another.[8] The indices $ n $ for which this expression yields a prime are cataloged in the On-Line Encyclopedia of Integer Sequences as A014545.[8] In contrast, the minus-one variant, $ p_n# - 1 $, These indices appear in OEIS sequence A057704.[9]

Historical context

Link to Euclid's infinitude proof

Euclid's theorem, dating to approximately 300 BCE, establishes the infinitude of prime numbers through a constructive proof that assumes a finite list of primes p1,p2,…,pkp_1, p_2, \dots, p_k and forms the number N=p1×p2×⋯×pk+1N = p_1 \times p_2 \times \dots \times p_k + 1, which equals the primorial pk#+1p_k\# + 1.[10] This NN, known as an Euclid number, cannot be divisible by any of the primes p1p_1 through pkp_k, so it either is prime itself or has a prime factor larger than pkp_k, thereby yielding a new prime not in the initial list.[11] Primorial primes of the form pn#+1p_n\# + 1—also known as Euclid primes—arise precisely when this Euclid number NN is prime, representing special cases where Euclid's construction directly produces a prime rather than a composite with novel factors.[11] Such instances confirm the proof's mechanism by providing explicit examples of primes beyond any finite enumeration.[11] An extension to the construction using minus one, forming pn#−1p_n\# - 1, yields another class of primorial primes, though this variant receives less emphasis in Euclid's classical argument, which relies on addition to ensure N>1N > 1.[12] In modern number theory, primorials underpin the generation of Euclid numbers, whose prime factors—whether the number itself is prime or composite—consistently introduce primes outside the initial set, reinforcing the proof's iterative extension to infinity.[13]

Early identifications and nomenclature

The construction of numbers as the product of known primes plus or minus one has ancient origins, serving as a key element in Euclid's proof of the infinitude of primes around 300 BCE, where such forms were used to generate new primes.[14] Small primorial primes, including 3, 5, 7, and 31, were recognized as primes in early number theory, with their structure as products of initial primes ±1 noted in classical and 19th-century works on prime forms, though without modern nomenclature.[1] The term "primorial" for the product of the first n primes was coined by Harvey Dubner in 1987, drawing an analogy to the factorial as a product of consecutive integers.[3] In the same publication, Dubner introduced the phrase "primorial primes" to describe primes of the form pn# ± 1, marking the formal nomenclature amid growing interest in computational prime hunting.[4] Systematic catalogs of primorial primes emerged in the 1990s through the Online Encyclopedia of Integer Sequences (OEIS), with sequences such as A018239 listing indices for pn# + 1 primes, facilitating further study.[15]

Small primorial primes

List of the first few

The smallest primorial primes arise from the first few values of pn#±1p_n\# \pm 1, where pn#p_n\# denotes the primorial (product of the first nn primes), along with the prime primorial 2 itself.[1] The following table enumerates these for n=1n = 1 to 1515, listing only the cases where pn#±1p_n\# \pm 1 is prime:
nnpn#p_n\#FormPrime
12+13
26+17
26-15
330+131
330-129
4210+1211
52310+12311
52310-12309
630030-130029
11200560490130+1200560490131
13304250263527210-1304250263527209
[1] Up to n=15n=15, 12 such primorial primes are known (including 2).[7] A partial list of the first few is 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209 (OEIS A228486).[7] Notably, 2309 (11#−111\# - 1) and 2311 (11#+111\# + 1) form a twin prime pair.[1]

Primality proofs for small cases

For the smallest primorial primes, such as those arising from the first few values of n≤5n \leq 5, primality can be verified using the basic method of trial division, which involves checking for divisibility by all prime numbers up to the square root of the candidate.[16] For instance, consider 31=5#+131 = 5\# + 1, where 5#=305\# = 30; since 31≈5.57\sqrt{31} \approx 5.57, it suffices to test divisibility by the primes 2, 3, and 5, none of which divide 31, confirming its primality.[17] Similarly, for 29=5#−129 = 5\# - 1, the same primes up to ⌊29⌋=5\lfloor \sqrt{29} \rfloor = 5 do not divide it.[17] This approach extends straightforwardly to slightly larger small cases within n≤5n \leq 5. For 211=7#+1211 = 7\# + 1 with 7#=2107\# = 210, 211≈14.52\sqrt{211} \approx 14.52, so trial division checks primes up to 13 (2, 3, 5, 7, 11, 13), revealing no divisors. For 2311=11#+12311 = 11\# + 1 where 11#=231011\# = 2310, 2311≈48.07\sqrt{2311} \approx 48.07, requiring checks against primes up to 47; exhaustive division shows none divide 2311.[18] The case of 2309=11#−12309 = 11\# - 1 follows analogously, with no prime divisors up to ⌊2309⌋=48\lfloor \sqrt{2309} \rfloor = 48.[19] These computations are feasible by hand or simple programs due to the limited range of trial divisors. Turning to n=6n=6, the primorial 13#=3003013\# = 30030 yields 30031=13#+130031 = 13\# + 1, which is composite with prime factorization 30031=59×50930031 = 59 \times 509.[20] This factorization demonstrates non-primality directly, as both 59 and 509 are primes greater than 1. In contrast, 30029=13#−130029 = 13\# - 1 is prime, verified by trial division against all primes up to ⌊30029⌋=173\lfloor \sqrt{30029} \rfloor = 173, with no divisors found.[21] For these semi-primorial forms, where the candidate p=pn#±1p = p_n\# \pm 1 has p−1p-1 or p+1p+1 fully factored as the primorial (a product of known distinct primes), Pocklington's theorem provides an alternative certification method even in small cases. The theorem states that if n−1=FRn-1 = FR with F>nF > \sqrt{n}, gcd⁡(F,R)=1\gcd(F, R) = 1, and for each prime qq dividing FF there exists an integer aa such that an−1≡1(modn)a^{n-1} \equiv 1 \pmod{n} and gcd⁡(a(n−1)/q−1,n)=1\gcd(a^{(n-1)/q} - 1, n) = 1, then nn is prime. Here, the complete factorization of the primorial satisfies the conditions efficiently, yielding a verifiable proof without full trial division.

Mathematical properties

Connections to number theory

Primorial primes play a significant role in recursive constructions for generating new primes, such as the Euclid-Mullin sequence, where each term is the least prime factor of one plus the product of all preceding terms in the sequence. This sequence begins with 2 and proceeds by taking the least prime factor of 1+∏k=1n−1ak1 + \prod_{k=1}^{n-1} a_k, yielding primes like 3, 7, 43, and 13, mirroring the primorial-based approach in Euclid's proof of the infinitude of primes by ensuring each new factor exceeds previous ones.[22] Although the Euclid-Mullin sequence diverges from standard primorials after initial terms, primorial primes exemplify how products of initial primes plus or minus one can produce novel primes in such iterative processes, extending the foundational idea of Euclid's construction to analytic number theory.[1] Heuristics derived from the prime number theorem suggest that there are infinitely many primorial primes, though this remains unproven. The primorial pn#p_n\# grows exponentially, with log⁡(pn#)∼pn∼nlog⁡n\log(p_n\#) \sim p_n \sim n \log n by the prime number theorem, implying that the probability of pn#±1p_n\# \pm 1 being prime is approximately 1/log⁡(pn#)∼1/(nlog⁡n)1 / \log(p_n\#) \sim 1/(n \log n). Since the harmonic series ∑1/(nlog⁡n)\sum 1/(n \log n) diverges, the expected number of such primes is infinite, supporting the conjecture despite the lack of a rigorous proof. This density heuristic aligns with broader probabilistic models in number theory, where the divergence indicates positive density in the logarithmic scale, but conditional results under assumptions like the Riemann hypothesis provide only partial bounds. Large primorial primes, particularly of the form pn#+1p_n\# + 1, imply the existence of substantial prime gaps immediately following them, as the subsequent integers pn#+kp_n\# + k for primes k=2,3,…,pnk = 2, 3, \dots, p_n are divisible by kk and thus composite, creating a gap of at least pn−1≈nlog⁡np_n - 1 \approx n \log n. This construction demonstrates arbitrarily large gaps in the primes, with the size scaling with the primorial's magnitude, and extends to multiples of the primorial where sieving by small primes enforces extended composite runs.[23] Such implications reinforce lower bounds on maximal prime gaps G(X)≫log⁡Xlog⁡log⁡X/log⁡log⁡log⁡XG(X) \gg \log X \log \log X / \log \log \log X in intervals up to XX, with primorial primes providing explicit examples of how prime density varies around highly composite numbers.[1]

Large known primorial primes

Records for p_n# - 1

The largest known primorial prime of the form pn#−1p_n\# - 1 as of November 2025 is 6533299#−16533299\# - 1, with 2,835,864 digits, discovered in August 2024 through the PrimeGrid distributed computing project.[24] This prime was rigorously verified as prime using the Brillhart-Lehmer-Selfridge method with an N+1 test, confirming its status after extensive computational testing.[24] Prior to this discovery, the record was held by 3267113#−13267113\# - 1, which has 1,418,398 digits and was found in September 2021, also by PrimeGrid participants.[4] These advancements highlight the ongoing efforts in distributed primality searches, where multiple large candidates have emerged in recent years due to improved sieving and proving techniques. The following table summarizes the five largest known primorial primes of this form, based on digit count:
RanknnFormDigitsDiscovery DateProject/Discoverer
165332996533299#−16533299\# - 12,835,864August 2024PrimeGrid (p447)
263549776354977#−16354977\# - 12,758,832August 2024PrimeGrid (p446)
347780274778027#−14778027\# - 12,073,926August 2024PrimeGrid (p442)
432671133267113#−13267113\# - 11,418,398September 2021PrimeGrid (p301)
510981331098133#−11098133\# - 1476,311March 2012PrimeGrid (p346)
All listed primes have been proven prime using advanced deterministic methods, such as the Brillhart-Lehmer-Selfridge N±1 tests, and are archived in the Prime Pages database.[4]

Records for p_n# + 1

The largest known primorial prime of the form pn#+1p_n\# + 1 as of November 2025 is 9562633#+19562633\# + 1, which has 4,151,498 digits and was discovered on June 29, 2025, by the PrimeGrid project.[25] This prime was initially identified as a probable prime (PRP) using probabilistic tests and subsequently verified deterministically via the N-1 test with the Brillhart-Lehmer-Selfridge method, trial division, and PRP checks, confirming its primality.[25] The previous record holder was 7351117#+17351117\# + 1, a 3,191,401-digit prime discovered on September 20, 2024, also by PrimeGrid, which underwent similar PRP screening followed by deterministic proof using the N-1 test.[26] These discoveries highlight the ongoing efforts in distributed computing to identify large primorial primes, with PrimeGrid coordinating searches for forms like pn#±1p_n\# \pm 1. The following table lists the five largest known primorial primes of the form pn#+1p_n\# + 1, where the value given (e.g., 9562633) denotes the largest prime factor pp in the primorial p#p\#:
RankPrimorial PrimeDigitsDiscovery DateDiscoverer (PrimeGrid Subproject)
19562633#+19562633\# + 14,151,498June 29, 2025p451
27351117#+17351117\# + 13,191,401September 20, 2024p448
36369619#+16369619\# + 12,765,105August 15, 2024p445
45256037#+15256037\# + 12,281,955July 27, 2024p444
54328927#+14328927\# + 11,878,843April 2024p442
Each of these primes was proven using a combination of PRP tests for initial detection and deterministic methods like the N-1 test for verification, ensuring reliability despite their immense size.[4][27][28][29]

Computational discovery

Search methods and algorithms

The search for primorial primes relies on efficient sieving to exclude composites with small factors and sophisticated primality testing to screen and verify candidates of the form pn#±1p_n\# \pm 1. Computations handle the immense scale of large primorials through multi-precision arithmetic, avoiding full explicit representation where possible by employing modular operations throughout.[30] Sieving begins by testing divisibility by primes q>pnq > p_n up to a practical limit, typically by computing pn#mod  qp_n\# \mod q via sequential multiplication of the first nn primes modulo qq and checking if the result equals ∓1\mp 1. This eliminates many composites early, with tools like mtsieve enabling multi-threaded processing for ranges of candidates. For numbers passing initial sieving but suspected to be composite, the elliptic curve method (ECM) is applied using GMP-ECM to detect factors of moderate size (up to 50-60 digits), often resolving pseudoprimes without exhaustive factorization.[30][31] Initial primality screening employs Lucas pseudoprime tests, particularly strong Lucas probable prime (PRP) tests based on Lucas sequences, which offer probabilistic certification with error rates below 4−t4^{-t} for tt iterations and are well-suited to general-form numbers like primorials. These are implemented in software such as PRST, running in hours on multi-core CPUs for candidates with hundreds of thousands of digits. Promising PRP candidates undergo further verification with deterministic PRP tests using multiple bases in OpenPFGW, providing practical certainty for numbers up to certain sizes.[30][32] For formal primality proofs, the elliptic curve primality proving (ECPP) algorithm is the standard, constructing a chain of elliptic curve witnesses that certify primality in expected subexponential time, outperforming alternatives for numbers exceeding 10,000 digits. Implementations like Primo generate verifiable certificates for primorial primes with up to millions of digits. The Adleman–Pomerance–Rumely (APRT) test serves as an alternative deterministic method, relying on properties of cyclotomic fields, though it is generally slower in practice for very large inputs.[33] Parallelization is critical for scalability, with multi-threading in tools like mtsieve and PRST distributing modular multiplications across CPU cores to accelerate sieving and PRP testing by factors of 4-64 depending on hardware. For extensive sieving over many moduli, GPU acceleration computes primorial products modulo small primes in parallel, exploiting the SIMD nature of graphics processors to process thousands of residues simultaneously. The core challenge lies in managing primorial sizes reaching 10610^6 digits or more, where full-number storage exceeds memory limits; thus, all operations—from sieving to exponentiations in PRP/ECPP—rely on modular arithmetic to represent and manipulate pn#±1p_n\# \pm 1 implicitly.[30][34]

Distributed projects and milestones

The PrimeGrid project, launched in 2005 as a volunteer computing initiative utilizing the BOINC platform, coordinates global efforts to discover large primes, including primorial primes of the form pn#±1p_n\# \pm 1.[35] Its Primorial Prime Search subproject, initiated in 2008, builds on a prior 2005 coordinated effort that tested up to n≈650,000n \approx 650,000 and employs distributed computing to sieve and test candidates for primality, primarily using probabilistic methods followed by verification.[36] Volunteers contribute idle CPU cycles from personal computers, enabling searches that individual efforts could not sustain due to the exponential growth in computational demands.[35] Key milestones in primorial prime discovery highlight the transition from individual computations to collaborative distributed projects. An early large primorial prime, 42209#+142209\# + 1 with 18,241 digits, was found through individual efforts in the late 20th century, representing one of the first significant advances beyond small cases.[4] The shift to distributed searching accelerated in the 2010s with PrimeGrid's involvement; for instance, in December 2010, the project discovered 843301#−1843301\# - 1, a 365,851-digit prime, marking a breakthrough in scale and confirming the efficacy of volunteer networks for such searches.[36] Subsequent efforts expanded the tested range dramatically, with sieving completed up to n=25×106n = 25 \times 10^6 by 2024.[37] As of 2025, PrimeGrid's searches continue beyond n=107n = 10^7, with the project progressing toward n=58×106n = 58 \times 10^6 at a rate of approximately 10,000 digits per day, though full completion to produce mega-digit candidates may require years of sustained effort.[37] with recent discoveries including four in August 2024 by PrimeGrid contributors, such as 6369619#+16369619\# + 1.[4] These findings contribute to records like the largest known pn#+1p_n\# + 1, 9562633#+19562633\# + 1 (4,151,498 digits, June 2025).[4] Distributed searches face significant challenges, including high energy costs that have contributed to declining volunteer participation since the early 2010s, as participants weigh electricity expenses against computational contributions.[35] Additionally, verifying probable primorial primes—often exceeding millions of digits—requires extensive time using deterministic methods like ECPP, with some validations taking weeks or months due to the numbers' size and complexity.[37]
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