# 13082761331670031 is a prime number

In number theory there is one Euclidean number a natural number of the form \ ({\ displaystyle E_ {n} = p_ {n} \ # + 1} \), where \ ({\ displaystyle p_ {n} \ # = 2 \ cdot 3 \ cdot 5 \ cdot 7 \ cdot \ ldots \ cdot p_ {n}} \) is the product of the first \ ({\ displaystyle n} \) prime numbers up to \ ({\ displaystyle p_ {n}} \) (prime faculty).

### Origin of name

These numbers were named after the ancient Greek mathematician Euclid, who was the first to prove in Euclid's theorem that there are infinitely many prime numbers. In doing so, he multiplied a set of prime numbers, added one to them and got a new number that none of the previous prime numbers could have as a divisor. So either this number was a prime number or it had prime divisors that did not appear in the previous prime number set. Euclidean numbers that are prime numbers are called Euclidean prime numbers (not all Euclidean numbers are prime numbers).

### Examples

• The first Euclidean number in literature is either \ ({\ displaystyle E_ {0} = p_ {0} \ # + 1 = 1 + 1 = 2} \) or \ ({\ displaystyle E_ {1} = p_ {1 } \ # + 1 = 2 + 1 = 3} \), depending on whether you define \ ({\ displaystyle p_ {0} \ #: = 1} \)[1] or not.[2]
• The first four prime numbers are \ ({\ displaystyle 2,3,5} \) and \ ({\ displaystyle 7} \). The product of these four prime numbers gives the prime faculty \ ({\ displaystyle p_ {4} \ # = 7 \ # = 2 \ times 3 \ times 5 \ times 7 = 210} \). So the Euclidean number is \ ({\ displaystyle E_ {4} = p_ {4} \ # + 1 = 7 \ # + 1 = 210 + 1 = 211} \).
• The first Euclidean numbers are (starting with \ ({\ displaystyle n = (0), 1,2, \ ldots} \)):
(2), 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, 7420738134811, 304,250,263,527,211, 13082761331670031, 614889782588491411, 32589158477190044731, 1922760350154212639071, 117288381359406970983271, 7858321551080267055879091 ... (sequence A006862 in OEIS)
• These Euclidean numbers have one or more prime factors. The following list gives the smallest of these prime factors for \ ({\ displaystyle E_ {n}} \) (with \ ({\ displaystyle n = (0), 1,2, \ ldots} \)):
(2), 3, 7, 31, 211, 2311, 59, 19, 347, 317, 331, 200560490131, 181, 61, 167, 953, 73, 277, 223, 54730729297, 1063, 2521, 22093, 265739, 131, 2336993, 960703, 2297, 149, 334507, 5122427, 1543, 1951, 881, 678279959005528882498681487, 87549524399, 23269086799180847, ... (episode A051342 in OEIS)
Example:
The list above shows that in the 7th position (without \ ({\ displaystyle (2)} \)) there is the number \ ({\ displaystyle 19} \). Thus the smallest divisor of \ ({\ displaystyle E_ {7} = 510511} \) is the number \ ({\ displaystyle 19} \).
• The following list gives the largest of these prime factors for \ ({\ displaystyle E_ {n}} \) (with \ ({\ displaystyle n = (0), 1,2, \ ldots} \)):
(2), 3, 7, 31, 211, 2311, 509, 277, 27953, 703763, 34231, 200,560,490,131, 676,421, 11,072,701, 78339888213593, 13808181181, 18564761860301, 19026377261, 525956867082542470777, 143,581,524,529,603, 2892214489673, 16156160491570418147806951, 96888414202798247, 1004988035964897329167431269, ... (Follow A002585 in OEIS)
Example:
The list above shows that the 7th position (without \ ({\ displaystyle (2)} \)) contains the number \ ({\ displaystyle 277} \). Thus the greatest divisor of \ ({\ displaystyle E_ {7} = 510511} \) is the number \ ({\ displaystyle 277} \).
• The following list gives the first \ ({\ displaystyle n} \) for which the Euclidean number \ ({\ displaystyle E_ {n}} \) is prime:
(0), 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... ( Episode A014545 in OEIS)
Example:
In the 6th position of the above list (without \ ({\ displaystyle (0)} \)) there is the number \ ({\ displaystyle 11} \). Thus \ ({\ displaystyle E_ {11} = p_ {11} \ # + 1 = 2 \ times 3 \ times 5 \ times 7 \ times 11 \ times 13 \ times 17 \ times 19 \ times 23 \ times 29 \ cdot 31 + 1 = 200560490131} \) the 6th Euclidean prime number.
• The largest known Euclidean prime number (as of July 8, 2018) is \ ({\ displaystyle E_ {33237} = p_ {33237} \ # + 1 = 392113 \ # + 1} \). It has \ ({\ displaystyle 169966} \) spots and was discovered by Daniel Heuer on September 20, 2001.[3]

### properties

• Not all Euclidean numbers are prime numbers.
Proof:
The sixth Euclidean number is already a composite number: \ ({\ displaystyle E_ {6} = p_ {6} \ # + 1 = 13 \ # + 1 = 2 \ times 3 \ times 5 \ times 7 \ times 11 \ times 13 + 1 = 30031 = 59 \ cdot 509} \). \ ({\ displaystyle \ Box} \)
Proof:
A counterexample is sufficient:
\ ({\ displaystyle E_ {8} = 510.511} \) and \ ({\ displaystyle E_ {18} = 1.922.760.350.154.212.639.071} \) have the greatest common factor \ ({\ displaystyle {\ operatorname {ggT } (E_ {8}, E_ {18})} = 277} \). \ ({\ displaystyle \ Box} \)
• Let \ ({\ displaystyle E_ {n}} \) be an arbitrary Euclidean number. Then the following applies:
\ ({\ displaystyle E_ {n} = 4k + 3} \) with \ ({\ displaystyle k \ in \ mathbb {N}} \)
In other words:
\ ({\ displaystyle E_ {n} \ equiv 3 {\ pmod {4}}} \)
Proof:
The product of odd (prime) numbers is again odd and, written with congruences, either \ ({\ displaystyle \ equiv 1 {\ pmod {4}}} \) or \ ({\ displaystyle \ equiv 3 {\ pmod {4}}} \). The prime faculty \ ({\ displaystyle p_ {n} \ #} \) is the product of \ ({\ displaystyle 2} \) and several odd prime numbers and thus either \ ({\ displaystyle \ equiv 2 \ cdot 1 \ equiv 2 {\ pmod {4}}} \) or \ ({\ displaystyle \ equiv 2 \ cdot 3 = 6 \ equiv 2 {\ pmod {4}}} \). So in both cases it is \ ({\ displaystyle \ equiv 2 {\ pmod {4}}} \). For a Euclidean number you have to add \ ({\ displaystyle 1} \) to the prime faculty and get \ ({\ displaystyle E_ {n} = p_ {n} \ # + 1 \ equiv 2 + 1 = 3 {\ pmod { 4}}} \) what was to be shown. \ ({\ displaystyle \ Box} \)
• Let \ ({\ displaystyle E_ {n}} \) be a Euclidean number with \ ({\ displaystyle n \ geq 3} \). Then the following applies:
The last digit (i.e. the units digit) of \ ({\ displaystyle E_ {n}} \) is always \ ({\ displaystyle 1} \).
In other words:
• \ ({\ displaystyle E_ {n} = 10k + 1} \) with \ ({\ displaystyle k \ in \ mathbb {N}} \) for \ ({\ displaystyle n \ geq 3} \)
• \ ({\ displaystyle E_ {n} \ equiv 1 {\ pmod {10}}} \) for \ ({\ displaystyle n \ geq 3} \)
Proof:
For \ ({\ displaystyle n \ geq 3} \) \ ({\ displaystyle E_ {n} -1 = p_ {n} \ # + 1-1 = 2 \ cdot 3 \ cdot 5 \ cdot \ ldots \ cdot p_ {n}} \). Thus \ ({\ displaystyle E_ {n} -1} \) is definitely through \ ({\ displaystyle 2} \) and \ ({\ displaystyle 5} \) and thus also through \ ({\ displaystyle 10} \) divisible. \ ({\ displaystyle E_ {n} -1} \) has a \ ({\ displaystyle 0} \) in the units place. If you add \ ({\ displaystyle 1} \) to this, you get a \ ({\ displaystyle 1} \) in the units. \ ({\ displaystyle \ Box} \)
• Let \ ({\ displaystyle E_ {n} = 2 \ cdot 3 \ cdot 5 \ cdot 7 \ cdot \ ldots \ cdot p_ {n} +1} \) be a Euclidean number. Then the following applies:
\ ({\ displaystyle E_ {n} \ equiv 1 {\ pmod {p}}} \) for all \ ({\ displaystyle p \ leq p_ {n}} \)
Proof:
The proof results from the definition of the Euclidean numbers. \ ({\ displaystyle E_ {n} = p_ {n} \ # + 1 = 2 \ times 3 \ times 5 \ times 7 \ times \ times \ times \ times p_ {n} + 1 = k \ times p + 1} \ ) with \ ({\ displaystyle k \ in \ mathbb {N}} \) and \ ({\ displaystyle p \ leq p_ {n}} \). Thus \ ({\ displaystyle E_ {n} = k \ cdot p + 1 \ equiv 1 {\ pmod {p}}} \) \ ({\ displaystyle \ Box} \)

### generalization

A Euclidean number of the 2nd kind (or Heartache number, named after Ernst Eduard Kummer[5][6]) is an integer of the form \ ({\ displaystyle {\ overline {E}} _ {n} = p_ {n} \ # - 1} \), where \ ({\ displaystyle p_ {n} \ # = 2 \ cdot 3 \ cdot 5 \ cdot 7 \ cdot \ ldots \ cdot p_ {n}} \) is the product of the first \ ({\ displaystyle n} \) prime numbers up to \ ({\ displaystyle p_ {n}} \) .

### Examples

• The first four prime numbers are \ ({\ displaystyle 2,3,5} \) and \ ({\ displaystyle 7} \). The product of these four prime numbers gives the prime faculty \ ({\ displaystyle p_ {4} \ # = 7 \ # = 2 \ times 3 \ times 5 \ times 7 = 210} \). Thus the fourth Euclidean number of the 2nd kind is the number \ ({\ displaystyle {\ overline {E}} _ {4} = p_ {4} \ # - 1 = 7 \ # - 1 = 210-1 = 209} \).
• The first Euclidean numbers of the 2nd kind are:
1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, 7420738134809, 304250263527209, 13082761331670029, 614889782588491409, 32589158477190044729, ...
• These Euclidean numbers of the 2nd kind have one or more prime factors. The following list gives the smallest of these prime factors for \ ({\ displaystyle {\ overline {E}} _ {n}} \) (with \ ({\ displaystyle n = 1,2, \ ldots} \)):
1, 5, 29, 11, 2309, 30029, 61, 53, 37, 79, 228737, 229, 304250263527209, 141269, 191, 87337, 27600124633, 1193, 163, 260681003321, 313, 163, 139, 23768741896345550770650537601358309, 66683309 2990092035859, 15649, 17515703, 719, 295201, 15098753, 10172884549, 20962699238647, 4871, 673, 311, 1409, 1291, 331, 1450184819, 23497, 711427, 521, 673, 519577, 182309, 83077, 56543 641, 349, 389, ... (sequence A057713 in OEIS)
Example:
The list above shows that the number \ ({\ displaystyle 61} \) is in the 7th position. Thus the smallest divisor of \ ({\ displaystyle {\ overline {E}} _ {7} = 510509} \) is the number \ ({\ displaystyle 61} \).
• The following list gives the largest of these prime factors for \ ({\ displaystyle {\ overline {E}} _ {n}} \) (with \ ({\ displaystyle n = 1,2, \ ldots} \)):
1, 5, 29, 19, 2309, 30029, 8369, 929, 46027, 81894851, 876817, 38669, 304250263527209, 92608862041, 59799107, 1143707681, 69664915493, 1146665184811, 17975352936245519, 21009252024 in OE
Example:
The list above shows that the number \ ({\ displaystyle 8369} \) is in the 7th position. Thus the greatest divisor of \ ({\ displaystyle {\ overline {E}} _ {7} = 510509} \) is the number \ ({\ displaystyle 8369} \).
• The following list gives the first \ ({\ displaystyle n} \) for which the Euclidean number of the 2nd kind \ ({\ displaystyle {\ overline {E}} _ {n}} \) is prime:
2, 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... (continuation A057704 in OEIS)
Example:
The number \ ({\ displaystyle 24} \) is in the 6th position of the list above. Thus \ ({\ displaystyle {\ overline {E}} _ {24} = 23,768,741,896,345,550,770,650,537,601,358,309} \) is the 6th Euclidean prime number of the 2nd kind.
• The largest known Euclidean prime number 2nd type (as of July 8, 2018) is \ ({\ displaystyle {\ overline {E}} _ {85586} = p_ {85586} \ # - 1 = 1098133 \ # - 1} \). It has \ ({\ displaystyle 476311} \) spots and was discovered by James P. Burt on February 28, 2012.[7][8]

### properties

• Not all Euclidean numbers of the 2nd kind are prime numbers.
Proof:
The fourth Euclidean number of the 2nd kind is already a composite number: \ ({\ displaystyle {\ overline {E}} _ {4} = p_ {4} \ # - 1 = 7 \ # - 1 = 2 \ cdot 3 \ times 5 \ times 7-1 = 209 = 11 \ times 19} \). \ ({\ displaystyle \ Box} \)
• Euclidean numbers of the 2nd kind are not coprime to one another.
Proof:
A counterexample is sufficient:
\ ({\ displaystyle {\ overline {E}} _ {19} = 7.858.321.551.080.267.055.879.089} \) and \ ({\ displaystyle {\ overline {E}} _ {22} = 3.217.644.767 .340.672.907.899.084.554.129} \) have the greatest common factor \ ({\ displaystyle {\ operatorname {ggT} ({\ overline {E}} _ {19}, {\ overline {E}} _ {22 })} = 163} \). \ ({\ displaystyle \ Box} \)
• Let \ ({\ displaystyle {\ overline {E}} _ {n}} \) be a Euclidean number of the 2nd kind with \ ({\ displaystyle n \ geq 3} \). Then the following applies:
The last digit (i.e. the units digit) of \ ({\ displaystyle {\ overline {E}} _ {n}} \) is always \ ({\ displaystyle 9} \).
In other words:
• \ ({\ displaystyle {\ overline {E}} _ {n} = 10k + 9} \) with \ ({\ displaystyle k \ in \ mathbb {N}} \) for \ ({\ displaystyle n \ geq 3 } \)
• \ ({\ displaystyle {\ overline {E}} _ {n} \ equiv 9 {\ pmod {10}}} \) for \ ({\ displaystyle n \ geq 3} \)
Proof: works analogously to the above proof for "normal" Euclidean numbers.
For \ ({\ displaystyle n \ geq 3} \) \ ({\ displaystyle {\ overline {E}} _ {n} + 1 = p_ {n} \ # - 1 + 1 = 2 \ cdot 3 \ cdot 5 \ cdot \ ldots \ cdot p_ {n}} \). Thus \ ({\ displaystyle {\ overline {E}} _ {n} +1} \) is definitely through \ ({\ displaystyle 2} \) and \ ({\ displaystyle 5} \) and thus also through \ ({\ displaystyle 10} \) divisible. \ ({\ displaystyle {\ overline {E}} _ {n} +1} \) has a \ ({\ displaystyle 0} \) in the units place. If you subtract \ ({\ displaystyle 1} \), you get a \ ({\ displaystyle 9} \) in the units place. \ ({\ displaystyle \ Box} \)
• Let \ ({\ displaystyle {\ overline {E}} _ {n} = 2 \ times 3 \ times 5 \ times 7 \ times \ ldots \ times \ times p_ {n} -1} \) a Euclidean number of the 2nd kind Then the following applies:[9]
\ ({\ displaystyle {\ overline {E}} _ {n} \ equiv -1 {\ pmod {p}}} \) for all \ ({\ displaystyle p \ leq p_ {n}} \)
Proof:
The proof results from the definition of the Euclidean numbers of the 2nd kind. \ ({\ Displaystyle {\ overline {E}} _ {n} = p_ {n} \ # - 1 = 2 \ cdot 3 \ cdot 5 \ cdot 7 \ cdot \ ldots \ cdot p_ {n} -1 = k \ cdot p-1} \) with \ ({\ displaystyle k \ in \ mathbb {N}} \) and \ ({\ displaystyle p \ leq p_ {n}} \). Thus \ ({\ displaystyle {\ overline {E}} _ {n} = k \ cdot p-1 \ equiv -1 {\ pmod {p}}} \) \ ({\ displaystyle \ Box} \)

### Unsolved problems

• Are there infinitely many Euclidean primes of the 2nd kind?

### Individual evidence

1. ↑ Neil Sloane: Euclid numbers: 1 + product of the first n primes. OEIS, accessed July 8, 2018.
2. ↑ Eric W. Weisstein: Euclid Number. Retrieved July 8, 2018.
3. ↑ 392113 # + 1 on Prime Pages
4. abNeil Sloane: Euclid numbers: 1 + product of the first n primes - Comments. OEIS, accessed July 8, 2018.
5. ↑ Ernst Eduard Kummer: New elementary proof of the theorem that the number of all prime numbers is infinite. In: Monthly report Preuss. Akad. D. Knowledge Berlin. 1878, pp. 777-778.
6. ↑ Neil Sloane: Kummer numbers: -1 + product of first n consecutive primes - References. OEIS, accessed July 8, 2018.
7. ↑ 1098133 # - 1 on Prime Pages
8. ↑ 1098133 # - 1 on primegrid.com (PDF)
9. ↑ Neil Sloane: Kummer numbers: -1 + product of first n consecutive primes - Comments. OEIS, accessed July 8, 2018.

Categories:Integer set | Prime number | Number theory

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