Cubic prime

In number theory , a cubic prime of the 1st kind (from the English cuban prime ) is a prime number that has the following form:

${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$with integer and . ${\displaystyle x=y+1}$ ${\displaystyle y>0}$

These kinds of cubic primes were first explored in 1912 by Allan Joseph Champneys Cunningham in the article On quasi-Mersennian numbers .

A cubic prime of the 2nd kind has the following form:

${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$with integer and . ${\displaystyle x=y+2}$ ${\displaystyle y>0}$

This type of cubic prime was also first explored by AJC Cunningham in 1923 in the article Binomial Factorizations .

The English name cuban prime comes from cube number , not from Cuba .

properties

• Every cubic prime of the first kind can be converted into the following forms:

• ${\displaystyle p=(y+1)^{3}-y^{3}}$
• ${\displaystyle p=3y^{2}+3y+1}$
• ${\displaystyle p=3x^{2}-3x+1}$

Proof of the 1st form:

A cubic prime of the first kind has the form with . Thus: ${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$ ${\displaystyle x=y+1}$

${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}={\frac {(y+1)^{3}-y^{3}}{(y+1)-y}}={\frac {(y+1)^{3}-y^{3}}{1}}=(y+1)^{3}-y^{3}}$. ${\displaystyle \Box }$

Proof of the 2nd form:

A cubic prime of the 1st kind, as shown above, has the form . Thus: ${\displaystyle p=(y+1)^{3}-y^{3}}$

${\displaystyle p=(y+1)^{3}-y^{3}=y^{3}+3y^{2}+3y+1-y^{3}=3y^{2}+3y+1}$. ${\displaystyle \Box }$

Proof of the 3rd form:

As shown above, a cubic prime of the 1st kind has the form with (i.e. with ). Thus: ${\displaystyle p=3y^{2}+3y+1}$ ${\displaystyle x=y+1}$ ${\displaystyle y=x-1}$

${\displaystyle p=3y^{2}+3y+1=3(x-1)^{2}+3(x-1)+1=3x^{2}-6x+3+3x-3+1=3x^{2}-3x+1}$. ${\displaystyle \Box }$

• Every cubic prime of the first kind is a centered hexagonal number .

Proof:

A cubic prime of the 1st kind, as shown above, has the form . Centered hexagon numbers have the shape . ${\displaystyle p=3x^{2}-3x+1}$ ${\displaystyle 3n^{2}-3n+1}$ ${\displaystyle \Box }$

• Every cubic prime of the 2nd kind can be converted into the following forms:

• ${\displaystyle p={\frac {(y+2)^{3}-y^{3}}{2}}}$
• ${\displaystyle p=3y^{2}+6y+4}$
• ${\displaystyle p=3x^{2}-6x+4}$
• ${\displaystyle p=3n^{2}+1}$with , ${\displaystyle n=y+1}$ ${\displaystyle n>1}$

Proof of the 1st form:

A cubic prime of the 2nd kind has the form with . Thus: ${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$ ${\displaystyle x=y+2}$

${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}={\frac {(y+2)^{3}-y^{3}}{(y+2)-y}}={\frac {(y+2)^{3}-y^{3}}{2}}}$. ${\displaystyle \Box }$

Proof of the 2nd form:

A cubic prime of the 2nd kind, as shown above, has the form . Thus: ${\displaystyle p={\frac {(y+2)^{3}-y^{3}}{2}}}$

${\displaystyle p={\frac {(y+2)^{3}-y^{3}}{2}}={\frac {y^{3}+6y^{2}+12y+8-y^{3}}{2}}={\frac {6y^{2}+12y+8}{2}}=3y^{2}+6y+4}$. ${\displaystyle \Box }$

Proof of the 3rd form:

A cubic prime of the 2nd kind has, as shown above, the form with (i.e. with ). Thus: ${\displaystyle p=3y^{2}+6y+4}$ ${\displaystyle x=y+2}$ ${\displaystyle y=x-2}$

${\displaystyle p=3y^{2}+6y+4=3(x-2)^{2}+6(x-2)+4=3x^{2}-12x+12+6x-12+4=3x^{2}-6x+4}$. ${\displaystyle \Box }$

Proof of the 4th form:

A cubic prime of the 2nd kind, as shown above, has the form . If one substitutes , one obtains: ${\displaystyle p=3y^{2}+6y+4}$ ${\displaystyle y:=n-1}$

${\displaystyle p=3y^{2}+6y+4=3(n-1)^{2}+6(n-1)+4=3n^{2}-6n+3+6n-6+4=3n^{2}+1}$. ${\displaystyle \Box }$

Examples

• The prime number can be represented as and is therefore a cubic prime number of the first kind. ${\displaystyle p=61}$ ${\displaystyle p={\frac {5^{3}-4^{3}}{5-4}}={\frac {125-64}{1}}=61}$
• The smallest cubic prime numbers of the first kind are:

7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, ... (sequence A002407 in OEIS )

• If one represents the cubic prime numbers of the 1st kind in the form , the first ones are the following: ${\displaystyle p=3x^{2}-3x+1}$ ${\displaystyle x}$

2, 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, 26, 28, 29, 31, 33, 35, 38, 39, 42, 43, 46, 49, 50, 53, 56, 59, 63, 64, 67, 68, 75, 81, 82, 87, 89, 91, 92, 94, 96, 106, 109, 120, 124, 126, 129, 130, 137, 141, 143, 148, 154, 157, 158, 159, 165, 166, 171, 172, ... (sequence A002504 in OEIS )

Example:

If you take the number at the 30th position from this list , you get , and actually the 30th cubic prime number is of the 1st kind, as you can see from the previous list. ${\displaystyle 63}$ ${\displaystyle p=3\cdot 63^{2}-3\cdot 63+1=11719}$ ${\displaystyle p=11719}$

• The number of cubic prime numbers of the 1st kind, which are smaller than , can be read from the following list for : ${\displaystyle 10^{n}}$ ${\displaystyle n=0,1,2,\ldots }$

0, 1, 4, 11, 28, 64, 173, 438, 1200, 3325, 9289, 26494, 76483, 221530, 645685, 1895983, 5593440, 16578830, 49347768, 147402214, 441641536, 1326941536, 3996900895, 12066234206, 36501753353 ... (Follow A113478 in OEIS )

Example:

In the list above you can find the number in the 5th position . This means that cubic primes of the 1st kind are smaller than . ${\displaystyle 28}$ ${\displaystyle 28}$ ${\displaystyle 10^{4}=10000}$

• The currently largest known cubic prime of the 1st kind is the following:

${\displaystyle p={\frac {(100000845^{4096}+1)^{3}-(100000845^{4096})^{3}}{100000845^{4096}[phone]^{4096}}}=(100000845^{4096}+1)^{3}-(100000845^{4096})^{3}=3\cdot 100000845^{8192}+3\cdot 100000845^{4096}+1}$

She has jobs and was discovered by Jens Kruse Andersen on January 7, 2006. ${\displaystyle 65537}$

• The smallest cubic prime numbers of the 2nd kind are:

13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249, 129793, 139969, ... (sequence A002648 in OEIS )

generalization

A generalized cubic prime has the following form:

${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$ with integer ${\displaystyle x>y>0}$

properties

• Any generalized cubic prime can be converted into the following forms:

• ${\displaystyle p=x^{2}+xy+y^{2}}$ with integer ${\displaystyle x>y>0}$
• ${\displaystyle p=6k+1}$with and ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle k>0}$

In other words:

${\displaystyle p\equiv 1{\pmod {6}}}$

Proof of the 1st form:

Because of the formula (see here ): ${\displaystyle a^{3}-b^{3}=(a-b)\cdot (a^{2}+ab+b^{2})}$

One can transform into . ${\displaystyle p}$ ${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}=x^{2}+xy+y^{2}}$ ${\displaystyle \Box }$

Proof of the 2nd form:

Be with and . Then is . If you calculate all variants for and through, you get the four remaining classes . Thus, the representations can assume or . The representations and are always composed and the representation is also composed except for . So only the representation remains. ${\displaystyle p=x^{2}+xy+y^{2}\equiv z{\pmod {6}}}$ ${\displaystyle x\equiv m{\pmod {6}}}$ ${\displaystyle y\equiv n{\pmod {6}}}$ ${\displaystyle p\equiv m^{2}+mn+n^{2}\equiv z{\pmod {6}}}$ ${\displaystyle m=0,1,\ldots 5}$ ${\displaystyle n=0,1,\ldots 5}$ ${\displaystyle p\equiv z\equiv 0,1,3,4{\pmod {6}}}$ ${\displaystyle p}$ ${\displaystyle 6k,6k+1,6k+3}$ ${\displaystyle 6k+4}$ ${\displaystyle 6k}$ ${\displaystyle 6k+4}$ ${\displaystyle 6k+3}$ ${\displaystyle p=3}$ ${\displaystyle p=6k+1}$ ${\displaystyle \Box }$

Examples

• The smallest generalized cubic prime numbers of the form are: ${\displaystyle p=x^{2}+xy+y^{2}}$

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, ... (sequence A007645 in OEIS )

Strictly speaking, the prime number does not belong to the above definition of generalized cubic prime numbers, because one can only obtain with and thus the original requirement is not fulfilled. For all other numbers with would be and therefore not a prime number. ${\displaystyle p=3}$ ${\displaystyle p=3}$ ${\displaystyle x=y=1}$ ${\displaystyle x>y}$ ${\displaystyle 0<x=y\not =1}$ ${\displaystyle p=x^{2}+xy+y^{2}=3x^{2}}$

Individual evidence

1. ↑ ^{a b} Cuban prime . In: PlanetMath . (English)
2. ^ Allan JC Cunningham: On quasi-Mersennian numbers. Messenger of Mathematics 41 , 1912, p. 144 , accessed July 7, 2018 .  
3. ↑ 3 • 100000845 ⁸¹⁹² + 3 • 100000845 ⁴⁰⁹⁶ + 1 on Prime Pages
4. ↑ Umesh P. Nair: Elementary results on the binary quadratic form a ^ 2 + ab + b ^ 2, Theorem 10. p. 4 , accessed July 7, 2018 . 

Web links

• Cuban prime . In: PlanetMath . (English)
• Eric W. Weisstein : Cuban Prime . In: MathWorld (English).
• Chris K. Caldwell: Cuban Prime. Prime Pages, accessed July 7, 2018 . 

swell

• AJC Cunningham: On Quasi-Mersennian Numbers . In: Messenger of Mathematics . tape 41 . England 1912, p. 119-146 .
• AJC Cunningham: Binomial Factorizations . London 1923.