Prime structures¶
Primality is one yes-or-no property. Prime structure asks what else a proven prime does: how its digits behave, how it sits beside other primes, how it factors in a larger ring, or what period its reciprocal has. These questions use different mathematics, and a prime may belong to many classes at once.
Numerisect proves the input prime before applying proof-oriented classifications. Digital operations, recurrences, modular orders, norms and every further primality decision are performed in PARI/GP. Python only validates, invokes GP and parses complete tagged output.
Families, properties and sequences are different things¶
A formula family specifies candidates, such as Mersenne numbers \(2^p-1\), Proth numbers \(k2^n+1\), or repunits \((b^n-1)/(b-1)\). Membership in the family need not imply primality.
A property class starts with a prime and asks a second exact question. A safe prime \(q\) has \((q-1)/2\) prime; a Sophie Germain prime \(p\) has \(2p+1\) prime; a Wieferich prime satisfies
Three further congruences of this kind are searched rather than merely tested. A Wall–Sun–Sun prime satisfies \(p^2 \mid F_{p-(5\mid p)}\), strengthening the entry-point divisibility \(p \mid F_{p-(5\mid p)}\) that holds for every prime \(p\neq 5\); none is known. A Wilson prime strengthens Wilson's theorem \((p-1)!\equiv-1\pmod p\) to the square, and a Wolstenholme prime strengthens \(H_{p-1}\equiv0\pmod{p^2}\) to \(H_{p-1}\equiv0\pmod{p^3}\), equivalently \(\binom{2p-1}{p-1}\equiv1\pmod{p^4}\). See the primality laboratory for the search and its bounds.
A sequence class asks whether the prime occurs in a defined recurrence or established catalogue. Exact recurrences can often settle membership. An open-ended or finite catalogue cannot prove non-membership beyond its known boundary, so the correct result is inconclusive.
This distinction explains the three-way classifier contract: match, definite non-match, or inconclusive. See the complete 56-class table.
Prime neighbors, tuples and the two meanings of “k-prime”¶
Twin, cousin and sexy primes have a prime partner at distance 2, 4 and 6 respectively. A prime triplet or quadruplet occupies an admissible pattern such as \(\{0,2,6\}\) or \(\{0,2,6,8\}\). More generally, offsets \(H=\{h_1,\ldots,h_k\}\) form a prime \(k\)-tuple at \(n\) when every \(n+h_i\) is prime.
“k-prime” also has a separate standard meaning: an integer \(n\) with exactly \(k\) prime factors counted with multiplicity, \(\Omega(n)=k\). Thus semiprimes are 2-primes in that terminology. Numerisect keeps the operations separate:
- Prime tuples searches offset patterns among primes.
- Compare \(\omega(n)\) and \(\Omega(n)\) studies distinct and repeated prime-factor counts of general integers.
Admissibility is necessary for a tuple pattern to recur: for every prime \(\ell\), the offsets must not cover all residue classes modulo \(\ell\). It is not a proof of any future tuple. Hardy–Littlewood estimates are therefore labelled predictions, while every reported tuple member is tested individually.
Decimal structure depends on the base¶
Circular, palindromic, emirp, truncatable, digitally delicate and non-insertable primes refer to a numeral representation, not to the abstract integer alone. Unless a tool says otherwise, Numerisect uses base 10.
For example, 197 is circular in base 10 because 197, 971 and 719 are prime. Rotations that introduce a leading zero are interpreted as integers after rotation, whereas a left-truncatable definition may explicitly reject embedded leading-zero cases. The exact convention is part of the class definition; changing it changes the class.
The absolute-prime search groups a decimal rotation orbit once. In this application that historical tool name means circular primes; it is not a claim that “absolute prime” has one universally standardized definition.
Reciprocal periods and full-reptend primes¶
Let \(p\) be prime and let the base \(b\) be coprime to \(p\). The repeating-period length of \(1/p\) in base \(b\) is
Lagrange's theorem gives \(\operatorname{ord}_p(b)\mid p-1\). The maximum possible period is therefore \(p-1\), attained exactly when \(b\) is a primitive root modulo \(p\). In base 10, such a prime is called full reptend. For \(p=7\), the order of 10 is 6 and
Primes dividing the base are the terminating exceptions: \(1/2=0.5\) and \(1/5=0.2\) in base 10. Numerisect reports period zero rather than forcing them into the repeating case.
Computing the exact order generally requires factoring \(p-1\). That factorization, not long division, can dominate a large analysis. The native reciprocal engine streams a complete finite expansion or one full repetend directly to the report; the 100,000-digit browser limit is only a preview cap. See Reciprocals of primes.
Gaussian and Eisenstein primes¶
The Gaussian integers \(\mathbb Z[i]\) use the norm
If both coordinates are nonzero, \(a+bi\) is Gaussian prime exactly when \(a^2+b^2\) is an ordinary rational prime. On an axis, \(a\) or \(bi\) is Gaussian prime exactly when the nonzero absolute coordinate is a rational prime congruent to 3 modulo 4. A rational prime \(p\equiv1\pmod4\) splits because it is a sum of two squares; \(p\equiv3\pmod4\) remains prime in \(\mathbb Z[i]\); and 2 ramifies.
For the Eisenstein integers \(\mathbb Z[\omega]\), where \(\omega^2+\omega+1=0\), the norm is
Away from the three unit axes, primality is decided by whether this norm is a rational prime. A rational prime \(p\equiv1\pmod3\) splits, \(p\equiv2\pmod3\) remains prime, and 3 ramifies. These are statements about irreducibility in different rings: the same rational integer can behave differently after the coefficient ring is enlarged.
PARI/GP computes the norm and the exact axis criterion. JavaScript only maps the returned lattice points to screen coordinates.
Perfect numbers and Mersenne primes¶
A positive integer is perfect when the sum of its proper divisors equals itself, or equivalently when \(\sigma(n)=2n\). The Euclid–Euler theorem classifies every even perfect number:
where \(2^p-1\) is prime. Numerisect therefore proves the Mersenne factor before calling the constructed integer perfect. It makes no claim about odd perfect numbers; whether any exist remains open.
For odd prime \(p\), every prime divisor \(q\) of \(M_p=2^p-1\) has
The mod-8 condition retains two of the four odd residue classes—one half of the \(q=2kp+1\) progression. Testing \(2^p\equiv1\pmod q\) proves divisibility without constructing \(M_p\). Exhausting a finite \(k\)-range proves only that no divisor was found in that range; Lucas–Lehmer answers the separate primality question. See Mersenne numbers.
The case \(p=2\) gives \(M_2=3\) and is the trivial exception to that divisor form.
Where this appears in Numerisect¶
| Topic | Tool | Native operation |
|---|---|---|
| 56 overlapping classes | Classify a prime | prime_classifier.gp; PARI isprime, recurrences, congruences and guarded searches |
| Decimal rotations | Find circular primes | prime_structures.gp; rigorous primality of every rotation |
| Prime constellations | Prime tuples, classifier | PARI forprime/isprime; primesieve for eligible 64-bit tuple counts |
| Factor multiplicity | Compare ω(n) and Ω(n) | PARI factor and exact exponent sums |
| Reciprocal periods | Analyze 1/p, Full-reptend primes | PARI znorder, znprimroot, isprime |
| Gaussian primes | Check a + bi, Find Gaussian primes | PARI exact norm and rational primality |
| Eisenstein primes | Eisenstein primes a + bω | PARI exact norm/axis criterion |
| Even perfect numbers | Generate even perfect numbers | PARI isprime and Euclid–Euler construction |
| Mersenne divisors | Mersenne trial factoring | PARI ispseudoprime, modular powering and exact divisibility congruence |
References¶
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, sections on periodic decimals, primitive roots and sums of two squares.
- K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, chapters on Gaussian and Eisenstein integers.
- P. Ribenboim, The New Book of Prime Number Records, chapters on special prime forms, decimal properties and prime constellations.
- L. E. Dickson, History of the Theory of Numbers, volume I, for Euclid–Euler perfect numbers and classical special-prime results.
- R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, for computational primality, orders and Mersenne-number methods.