Smarandache Wellin Number

In number theory , a Smarandache-Wellin number is an integer whose digits in the decimal system (or any other number system ) consist of the concatenation of the first prime numbers (in this number system). ${\displaystyle n\in \mathbb {N} }$

For example, the first five prime numbers in the decimal system are 2, 3, 5, 7, and 11. So the fifth Smarandache – Wellin number is the number . ${\displaystyle n=235711}$

If a Smarandache-Wellin number is a prime number, it is called a Smarandache-Wellin prime number .

These numbers were named after the artist Florentin Smarandache and the mathematician Paul R. Wellin .

Examples

• The first Smarandache – Wellin numbers in the decimal system are as follows:

  2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, 235711131719232931, 23571113171923293137, ... (episode A019518 in OEIS )

• The first Smarandache – Wellin prime numbers in the decimal system are the following:

  2, 23, 2357, ... (sequence A069151 in OEIS )

The fourth Smarandache – Wellin prime already has 355 digits, the fifth 499 digits.

• The following list indicates how many Smarandache-Wellin numbers the respective Smarandache-Wellin prime numbers are:

  1, 2, 4, 128, 174, 342, 435, 1429 (episode A046035 in OEIS )

Example 1:

The third place in the list above is the number 4. Thus, the number which is obtained by concatenation of the first 4 primes itself is a prime number: . ${\displaystyle p=2357\in \mathbb {P} }$

Example 2:

The number 128 can be found in the fourth position in the list above. Thus, the number which is obtained by concatenation of the first 128 prime numbers, even a prime number . From this number you can see that the penultimate prime number (the 127th prime number) is the number 709 and the last (the 128th) prime number must be the number 719. This leads to the following list: ${\displaystyle p=23571113\ldots 709719\in \mathbb {P} }$

• The following list indicates with which prime number the Smarandache – Wellin prime numbers end:

2, 3, 7, 719, 1033, 2297, 3037, 11927 (episode A046284 in OEIS )

• The following list gives the number of digits of the first Smarandache – Wellin prime numbers:

1, 2, 4, 355, 499, 1171, 1543, 5719 (follow A263959 in OEIS )

Example:

The eighth position of the three lists above shows the numbers 1429, 11927 and 5719. Thus the eighth Smarandache-Wellin prime number is the concatenation of the first 1429 prime numbers and ends with this 1429th prime number, which is the prime number . This 1429 is Smarandache-Wellin number one 5719-digit PRP-number , which is a probable prime (that is, that it is still uncertain whether it is prime or maybe actually only a pseudo-prime is) It was founded by Eric W . Weisstein discovered in 1998. ${\displaystyle p=11927}$

• The next, i.e. the ninth Smarandache-Wellin prime number (if it exists) is at least the 34736th Smarandache-Wellin number, i.e. the concatenation of the first 34736 prime numbers.

Smarandache numbers and Champernowne numbers

In number theory , a Smarandache number is an integer whose digits in the decimal system (or any other number system ) consist of the concatenation of the first numbers (in this number system). The -th Smarandache number is abbreviated with . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$ ${\displaystyle Sm(n)}$

For example, the first five numbers in the decimal system are the numbers 1, 2, 3, 4, and 5. So the fifth Smarandache number is the number . ${\displaystyle Sm(5)=12345}$

If a Smarandache number is a prime number, it is called a Smarandache prime number . But none is known yet.

If you can break off in the middle of a Smarandache number, the resulting number is called the Champernowne number . For example, the twelfth Smarandache number is the number . The last two digits come from the number . If you break this number at the very end between and , you get the Champernowne number . ${\displaystyle Sm(12)=123456789101112}$ ${\displaystyle 12}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle m=12345678910111}$

If a Champernowne number is prime, it is called a Champernowne prime number .

The following number is called the Champernowne constant or, as above, the Champernowne number :

${\displaystyle C=0,1234567891011121314\ldots }$(Follow A033307 in OEIS )

These numbers were named after the mathematician David Gawen Champernowne ( en ).

Examples

• The first Smarandache numbers are the following:

1, 12, 123, 1234, 12345, 123456 , 1234567, 12345678, 123456789, 12345678910, 1234567891011, 123456789101112, ... ( continuation A007908 in OEIS )

• The number of digits in the first Smarandache numbers are as follows:

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, ... (episode A058183 in OEIS )

• The first Smarandache numbers in the dual system are the following:

0, 1, 110, 11011, 11011100, 11011100101, 11011100101110, 11011100101110111, 110111001011101111000, 1101110010111011110001001, ... (sequence A058935 in OEIS )

• The first Champernowne numbers are the following:

1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567891, 12345678910, 123456789101, 1234567891011, 12345678910111, 123456789101112, 1234567891011121, ... ( continuation A252043 in OEIS )

• The first Champernowne primes are as follows:

1234567891, 12345678910111, 123456789101112131415161, ... (Follow A176942 in OEIS )

• The number of digits of the first Champernowne prime numbers are as follows:

10, 14, 24, 235, 2804, 4347, 37735,… (Follow A071620 in OEIS )

The eighth (not yet discovered) Champernowne prime will have more than 37,800 digits.

Factoring Smarandache Numbers

The following table gives the prime factors of the first 30 Smarandache numbers.

${\displaystyle n}$ Factorization of ${\displaystyle Sm(n)}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 2^{2}\cdot 3}$ ${\displaystyle 3}$ ${\displaystyle 3\cdot 41}$ ${\displaystyle 4}$ ${\displaystyle 2\cdot 617}$ ${\displaystyle 5}$ ${\displaystyle 3\cdot 5\cdot 823}$ ${\displaystyle 6}$ ${\displaystyle 2^{6}\cdot 3\cdot 643}$ ${\displaystyle 7}$ ${\displaystyle 127\cdot 9721}$ ${\displaystyle 8}$ ${\displaystyle 2\cdot 3^{2}\cdot 47\cdot 14593}$ ${\displaystyle 9}$ ${\displaystyle 3^{2}\cdot 3607\cdot 3803}$ ${\displaystyle 10}$ ${\displaystyle 2\cdot 5\cdot 1234567891}$ ${\displaystyle 11}$ ${\displaystyle 3\cdot 7\cdot 13\cdot 67\cdot 107\cdot 630803}$ ${\displaystyle 12}$ ${\displaystyle 2^{3}\cdot 3\cdot 2437\cdot 2110805449}$ ${\displaystyle 13}$ ${\displaystyle 113\cdot 125693\cdot 869211457}$ ${\displaystyle 14}$ ${\displaystyle 2\cdot 3\cdot 205761315168520219}$ ${\displaystyle 15}$ ${\displaystyle 3\cdot 5\cdot 8230452606740808761}$

${\displaystyle n}$ Factorization of ${\displaystyle Sm(n)}$ ${\displaystyle 16}$ ${\displaystyle 2^{2}\cdot 2507191691\cdot 1231026625769}$ ${\displaystyle 17}$ ${\displaystyle 3^{2}\cdot 47\cdot 4993\cdot 584538396786764503}$ ${\displaystyle 18}$ ${\displaystyle 2\cdot 3^{2}\cdot 97\cdot 88241\cdot 801309546900123763}$ ${\displaystyle 19}$ ${\displaystyle 13\cdot 43\cdot 79\cdot 281\cdot 1193\cdot 833929457045867563}$ ${\displaystyle 20}$ ${\displaystyle 2^{5}\cdot 3\cdot 5\cdot 323339\cdot 3347983\cdot 2375923237887317}$ ${\displaystyle 21}$ ${\displaystyle 3\cdot 17\cdot 37\cdot 43\cdot 103\cdot 131\cdot 140453\cdot 802851238177109689}$ ${\displaystyle 22}$ ${\displaystyle 2\cdot 7\cdot 1427\cdot 3169\cdot 85829\cdot 2271991367799686681549}$ ${\displaystyle 23}$ ${\displaystyle 3\cdot 41\cdot 769\cdot 13052194181136110820214375991629}$ ${\displaystyle 24}$ ${\displaystyle 2^{2}\cdot 3\cdot 7\cdot 978770977394515241\cdot 1501601205715706321}$ ${\displaystyle 25}$ ${\displaystyle 5^{2}\cdot 15461\cdot 31309647077\cdot 1020138683879280489689401}$ ${\displaystyle 26}$ ${\displaystyle 2\cdot 3^{4}\cdot 21347\cdot 2345807\cdot 982658598563\cdot 154870313069150249}$ ${\displaystyle 27}$ ${\displaystyle 3^{3}\cdot 19^{2}\cdot 4547\cdot 68891\cdot 40434918154163992944412000742833}$ ${\displaystyle 28}$ ${\displaystyle 2^{3}\cdot 47\cdot 409\cdot 416603295903037\cdot 192699737522238137890605091}$ ${\displaystyle 29}$ ${\displaystyle 3\cdot 859\cdot 24526282862310130729\cdot 19532994432886141889218213}$ ${\displaystyle 30}$ ${\displaystyle 2\cdot 3\cdot 5\cdot 13\cdot 49269439\cdot 370677592383442753\cdot 17333107067824345178861}$

Generalizations

Because there are no known Smarandache prime numbers, generalizations are sought.

If you write consecutive numbers one after the other, but not necessarily with , but also with or etc., you get prime numbers. What is the smallest prime number that can be generated if you start with ? The following list provides information (if there is no known prime, 0 is given): ${\displaystyle k}$ ${\displaystyle n=1}$ ${\displaystyle n=2}$ ${\displaystyle n=3}$ ${\displaystyle n=1,2,3,\ldots }$

0 , 1, 1, 4, 1, 2, 1, 2, 179, 0 , 1, 2, 1, 4, 5, 28, 1, 3590, 1, 4, 0 , 0 , 1, 0 , 25, 122, 0 , 46, 1, 0 , 1, 0 , 71, 4, 569, 2, 1, 20, 5, 0 , 1, 2, 1, 8, 0 , 0 , 1, 0 , 193, 2, 0 , 0 , 1, 0 , 0 , 2, 5, 4, 1, 0 , 1, 2, 0 , 4, 5, 938, 1, 2, 119, 58, 1, 116, 1, 0 , 125, 346, 5, 2, 1, 2, 0 , 0 , 1, 0 , 0 , 32, ... (sequence A244424 in OEIS )

Example 1: In the 1st position of the above list there is a . ${\displaystyle 0}$

So at the moment no prime number is known that starts with . ${\displaystyle p=1\mathbf {2} 3\mathbf {4} 5\ldots }$

Example 2: In the 15th position in the list above there is a . ${\displaystyle 5}$

Thus the number is the smallest prime number that starts with and continues with the following numbers. ${\displaystyle p=15\mathbf {16} 17\mathbf {18} 19\in \mathbb {P} }$ ${\displaystyle 15}$

Example 3: In the 18th position in the list above is the number . ${\displaystyle 3590}$

Thus the number is the smallest prime number that starts with and continues with the following numbers. As you can see, it ends with a number and has digits. ${\displaystyle p=18\mathbf {19} 20\mathbf {21} 22\ldots 3607\in \mathbb {P} }$ ${\displaystyle 18}$ ${\displaystyle 3607}$ ${\displaystyle 13296}$

Example 4: In the 21st position of the list above there is a . ${\displaystyle 0}$

So at the moment no prime number is known that starts with . ${\displaystyle p=21\mathbf {22} 23\mathbf {24} 25\ldots }$

Let the number that starts with contain the decimal numbers 1, 2, 3, ..., k, but which is missing the nth number (for example is ). Then the smallest for which is a prime are the following (if there is no known prime, 0 is given): ${\displaystyle S(n,k)}$ ${\displaystyle 123456\ldots }$ ${\displaystyle S(4,9)=12356789}$ ${\displaystyle k}$ ${\displaystyle S(n,k)}$

2, 3, 7, 9, 11, 7, 11, 1873, 19, 14513, 13, 961, 0 , 653, 0 , 5109, 493, 757, 29, 1313, ... ( continuation A262300 in OEIS )

Example 1: At the . In the list above there is a . ${\displaystyle 3}$ ${\displaystyle 7}$

Thus the number is the smallest prime number that is missing, but that otherwise contains all numbers . ${\displaystyle S(3,7)=p=124567\in \mathbb {P} }$ ${\displaystyle 3}$ ${\displaystyle 1}$

Example 2: At the . The number is in the above list . ${\displaystyle 9}$ ${\displaystyle 19}$

Thus the number is the smallest prime number that is missing, but that otherwise contains all numbers . ${\displaystyle S(9,19)=p=1234567810111213141516171819\in \mathbb {P} }$ ${\displaystyle 9}$ ${\displaystyle 1}$

Example 3: At the . The number is in the above list . ${\displaystyle 13}$ ${\displaystyle 0}$

Thus no number of the form that is prime is known. ${\displaystyle S(13,k)=123456789101112141516171819\ldots k}$

Unsolved problems

It is assumed that there are an infinite number of Smarandache prime numbers, but not a single one has yet been found (as of December 2016). There are definitely no Smarandache prime numbers among the first 344,869 Smarandache numbers.

Web links

• Eric W. Weisstein : Smarandache-Wellin Number . In: MathWorld (English).
• Eric W. Weisstein : Smarandache-Wellin Prime . In: MathWorld (English).
• Eric W. Weisstein : Smarandache Number . In: MathWorld (English).
• Eric W. Weisstein : Smarandache Prime . In: MathWorld (English).
• Eric W. Weisstein : Champernowne Constant . In: MathWorld (English).

Individual evidence

1. ^ Neil Sloane : Numbers n such that the concatenation of the first n primes is a prime - Comments. OEIS , accessed August 3, 2018 . 
2. ↑ ^{a b} Eric W. Weisstein : Smarandache Prime . In: MathWorld (English).
3. ↑ Neil Sloane : Champernowne primes - Comments. OEIS , accessed August 3, 2018 .