Star prime

In number theory , a star prime number (from the English star prime ) is a prime number that cannot be represented as the sum of a smaller prime number and the double of a square of an integer . ${\displaystyle p}$ ${\displaystyle b\not =0}$

In other words: If there is no smaller prime number and no whole number for a prime number , so that is true, then one calls a star prime number. ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle b\not =0}$ ${\displaystyle p=q+2b^{2}}$ ${\displaystyle p}$

Rephrased a little bit you get: A prime number is called a star prime number, if no prime number results for all integer numbers . ${\displaystyle p}$ ${\displaystyle p-2b^{2}\not \in \mathbb {P} }$ ${\displaystyle b\not =0}$

These numbers were first mentioned on November 18, 1752 by Christian Goldbach in a letter to Leonhard Euler ( at the time he assumed that every odd whole number had the form with an integer and prime ) and about a century later, in 1856, by the German mathematician Moritz Stern , after whom these numbers were named. ${\displaystyle q+2b^{2}}$ ${\displaystyle b}$ ${\displaystyle q}$

Examples

• Be . Then you can subtract the first double square numbers from this prime number and check whether you get a prime number : ${\displaystyle p=137}$ ${\displaystyle p}$ ${\displaystyle 2b^{2}}$ ${\displaystyle q}$

${\displaystyle 137-2\cdot 1^{2}=135=3^{3}\cdot 5\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 2^{2}=129=3\cdot 43\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 3^{2}=119=7\cdot 17\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 4^{2}=105=3\cdot 5\cdot 7\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 5^{2}=87=3\cdot 29\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 6^{2}=65=5\cdot 13\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 7^{2}=39=3\cdot 13\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 8^{2}=9=3^{2}\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 137-2\cdot 9^{2}=-25<0}$

Obviously there isn't one such that is prime. Thus is a star prime. ${\displaystyle b>0}$ ${\displaystyle 137-2\cdot b^{2}\in \mathbb {P} }$ ${\displaystyle p=137}$

• Be . Again you check whether you get a prime number with the above procedure : ${\displaystyle p=97}$ ${\displaystyle q}$

${\displaystyle 97-2\cdot 1^{2}=95=5\cdot 19\not \in \mathbb {P} }$ is not a prime number.

${\displaystyle 97-2\cdot 2^{2}=89\in \mathbb {P} }$ is a prime number.

You can interrupt the calculation because you found a prime number and one such that is a prime number. So is not a star prime. The prime number calculated in this way is in this case not the only prime number that can be obtained in this way. Likewise also yields and prime. So there are three possibilities for obtaining a prime number with . These representations are called Goldbach representations of . ${\displaystyle q=89}$ ${\displaystyle b=2\not =0}$ ${\displaystyle 97-2\cdot b^{2}=q\in \mathbb {P} }$ ${\displaystyle p=97}$ ${\displaystyle q=89}$ ${\displaystyle 97-2\cdot 3^{2}=79\in \mathbb {P} }$ ${\displaystyle 97-2\cdot 5^{2}=47\in \mathbb {P} }$ ${\displaystyle p=97}$ ${\displaystyle p-2\cdot b^{2}}$ ${\displaystyle p=97}$

• The only known star primes are the following:

2, 3, 17, 137, 227, 977, 1187, 1493 (episode A042978 in OEIS )

There are no more star primes until . It is not known whether there are any larger ones. ${\displaystyle 2\cdot 10^{13}}$

• The following list gives all known odd numbers , not necessarily prime numbers, which have no Goldbach representations, which are therefore not of the form with prime : ${\displaystyle n}$ ${\displaystyle n=q+2b^{2}}$ ${\displaystyle q}$

1, 3, 17, 137, 227, 977, 1187, 1493, 5777, 5993 (episode A060003 in OEIS )

These numbers are called star numbers . Only two of these numbers are non-prime, namely 5777 and 5993.

• As mentioned above, a number often has several Goldbach representations. The following list gives the smallest number of the Goldbach representations has (with ascending , where also and is allowed): ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle k=1,2,3,\ldots }$ ${\displaystyle p=1}$ ${\displaystyle b=0}$

1, 3, 13, 19, 55, 61, 139, 139, 181, 181, 391, 439, 559, 619, 619, 829, 859, 1069, 1081, 1459, 1489, 1609, 1741, 1951, 2029, 2341, 2341, 3331, 3331, 3331, 3961, 4189, 4189, 4261, 4801, 4801, 5911, 5911, 5911, 6319, 6319, 6319, 8251, 8251, 8251, 8251, 8251 (sequence A007697 in OEIS )

Example:

The seventh and eighth position in the list above is the number . In fact, there are eight different (and thus seven different) Goldbach representations for this number (in this case a prime number), more than any other smaller number before (up to this number had the record with six Goldbach representations): ${\displaystyle n=139}$ ${\displaystyle n=61}$

${\displaystyle 139=139+2\cdot 0^{2}}$with , but strictly speaking is not allowed according to the definition of star primes because ${\displaystyle 139\in \mathbb {P} }$ ${\displaystyle b=0}$

${\displaystyle 139=137+2\cdot 1^{2}}$ With ${\displaystyle 137\in \mathbb {P} }$

${\displaystyle 139=131+2\cdot 2^{2}}$ With ${\displaystyle 131\in \mathbb {P} }$

${\displaystyle 139=121+2\cdot 3^{2}}$with , so no Goldbach representation ${\displaystyle 121=11^{2}\not \in \mathbb {P} }$

${\displaystyle 139=107+2\cdot 4^{2}}$ With ${\displaystyle 107\in \mathbb {P} }$

${\displaystyle 139=89+2\cdot 5^{2}}$ With ${\displaystyle 89\in \mathbb {P} }$

${\displaystyle 139=67+2\cdot 6^{2}}$ With ${\displaystyle 67\in \mathbb {P} }$

${\displaystyle 139=41+2\cdot 7^{2}}$ With ${\displaystyle 41\in \mathbb {P} }$

${\displaystyle 139=11+2\cdot 8^{2}}$ With ${\displaystyle 11\in \mathbb {P} }$

useful information

• With prime twins , the larger of the two prime numbers has the Goldbach representation . ${\displaystyle (p,p+2)}$ ${\displaystyle p+2=p+2\cdot 1^{2}}$
• For prime quadruplets , the largest of these four prime numbers has the Goldbach representation . ${\displaystyle (p,p+2,p+6,p+8)}$ ${\displaystyle p+8=p+2\cdot 2^{2}}$
• Leonhard Euler already suspected that the larger a prime number , the more (Goldbach) representations of the form there are for this number. That is why he was of the opinion that the above (short) list of the 8 star primes are all star primes that exist. ${\displaystyle p}$ ${\displaystyle p=q+2b^{2}}$
• In his letter to Leonhard Euler, Goldbach assumed that every odd whole number can be written in the form with prime or and and cited a representation of the form as an example for the star prime number . In doing so, he found representations of the form for all other prime numbers , which, however, do not correspond to the current definition of star prime numbers, because it is now required. In this respect, he claimed that all star numbers (with today's definition) are prime numbers. In the meantime, however, two (odd) star numbers are known that are not prime numbers, namely and , which definitely have no representation of the form . So Goldbach was wrong. ${\displaystyle n}$ ${\displaystyle n=q+2b^{2}}$ ${\displaystyle q\in \mathbb {P} }$ ${\displaystyle q=1}$ ${\displaystyle a\geq 0}$ ${\displaystyle p=17}$ ${\displaystyle 17=17+2\cdot 0^{2}}$ ${\displaystyle p=p+2\cdot 0^{2}}$ ${\displaystyle b\not =0}$ ${\displaystyle 5777=53\cdot 109}$ ${\displaystyle 5993=13\cdot 461}$ ${\displaystyle q+2b^{2}}$
• From 1856 Moritz Stern and his students examined all odd numbers up to and also found the two star numbers and , which are not prime numbers. However, he listed the prime number as the smallest star prime number and not the actually smallest odd star prime number . The reason for this is that at that time many mathematicians still regarded the number as a prime number, which is why it was not considered a star prime number because this number has the representation . ${\displaystyle 9000}$ ${\displaystyle 5777}$ ${\displaystyle 5993}$ ${\displaystyle p=17}$ ${\displaystyle p=3}$ ${\displaystyle 1}$ ${\displaystyle p=3}$ ${\displaystyle 3=1+2\cdot 1^{2}}$

Web links

• Star prime . In: PlanetMath . (English)

Individual evidence

1. ↑ ^{a b} Comments and links to OEIS A042978
2. ↑ ^{a b c d} Laurent Hodges: A lesser-known Goldbach conjecture
3. ↑ Toying with a lesser known Goldbach Conjecture ...
4. ↑ Chris K. Caldwell , Angela Reddick, Yeng Xiong: The History of the Primality of One: A Selection of Sources. Journal of Integer Sequences 15 , Article 12.9.8, 2012, pp. 1-40 , accessed on February 10, 2020 .