Cuban Prime
A Cuban prime is a prime number obtained from particular expressions involving the difference of two cubes. The name has nothing to do with Cuba; it comes from the word cube, because the defining expressions involve third powers. The Wikipedia article describes two main families. In the first, one considers consecutive integers, $x=y+1$, and forms
$
p=\frac{x^3-y^3}{x-y}.
$
Since $\frac{x^3-y^3}{x-y}=x^2+xy+y^2$, substituting $x=y+1$ gives $p=3y^2+3y+1$. Whenever this number is prime, it is called a Cuban prime of the first kind. The sequence begins $7,19,37,61,127,271,\ldots$. Interestingly, numbers of the form $3y^2+3y+1$ are exactly the centered hexagonal numbers, giving these primes a geometric interpretation as well.
The second family arises when the two integers differ by two, so $x=y+2$. The same quotient becomes $p=3y^2+6y+4$. Making the substitution $y=n-1$ gives the particularly simple expression $p=3n^2+1$, with $n>1$. Prime values of this polynomial form the second Cuban-prime sequence, beginning $13,109,193,433,769,1201,\ldots$. Thus Cuban primes provide an elementary example of an important theme in number theory: studying when polynomial expressions take prime values. Although the formulas are simple, determining which inputs actually produce primes becomes increasingly difficult as the numbers grow.
The first family also produces extraordinarily large primes. Wikipedia reports that, as of July 2023, the largest known Cuban prime had 3,153,105 decimal digits, obtained using $y=3^{3304301}-1$. This statement is explicitly dated in the article, so it should not necessarily be interpreted as the current 2026 record. The subject connects elementary algebra—the factorization $x^3-y^3=(x-y)(x^2+xy+y^2)$—with prime-number theory, polynomial prime generation, and figurate numbers.
Key Takeaways
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A Cuban prime is a prime number obtained from particular expressions involving the difference of two cubes. The name has nothing to do with Cuba; it comes from the word cube, because the defining expressions involve third powers. The Wikipedia article describes two main families. In the first, one considers consecutive integers, $x=y+1$, and forms
$
p=\frac{x^3-y^3}{x-y}.
$
Since $\frac{x^3-y^3}{x-y}=x^2+xy+y^2$, substituting $x=y+1$ gives $p=3y^2+3y+1$. Whenever this number is prime, it is called a Cuban prime of the first kind. The sequence begins $7,19,37,61,127,271,\ldots$. Interestingly, numbers of the form $3y^2+3y+1$ are exactly the centered hexagonal numbers, giving these primes a geometric interpretation as well.
The second family arises when the two integers differ by two, so $x=y+2$. The same quotient becomes $p=3y^2+6y+4$. Making the substitution $y=n-1$ gives the particularly simple expression $p=3n^2+1$, with $n>1$. Prime values of this polynomial form the second Cuban-prime sequence, beginning $13,109,193,433,769,1201,\ldots$. Thus Cuban primes provide an elementary example of an important theme in number theory: studying when polynomial expressions take prime values. Although the formulas are simple, determining which inputs actually produce primes becomes increasingly difficult as the numbers grow.
The first family also produces extraordinarily large primes. Wikipedia reports that, as of July 2023, the largest known Cuban prime had 3,153,105 decimal digits, obtained using $y=3^{3304301}-1$. This statement is explicitly dated in the article, so it should not necessarily be interpreted as the current 2026 record. The subject connects elementary algebra—the factorization $x^3-y^3=(x-y)(x^2+xy+y^2)$—with prime-number theory, polynomial prime generation, and figurate numbers.
Key Takeaways
- Definition: Cuban primes are primes generated from quotients of differences of cubes, especially when $x-y=1$ or $x-y=2$.
- Two principal forms: the families reduce to $3y^2+3y+1$ and $3n^2+1$.
- Geometric connection: Cuban primes of the first kind are prime centered hexagonal numbers.
- Significance: They illustrate how very simple polynomial formulas can generate primes while making the question of when they are prime a nontrivial number-theoretic problem.
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