CAVEAT: This site is under construction so please bear with any redundancy or imprecision. Constructive criticism only to wkehowski@cox.net or walter.kehowski@gcmail.maricopa.edu.

It seems that D-numbers as I call them are something new. This page is a repository of all known examples of D-numbers.
p, q, r, s always denote odd primes, and integers will always be positive.

Terminology: I am well aware of the terms "semiprime", "3-almost-prime" and so on, but these terms strike me as wrong or cumbersome. For example, 'semi' means 'half' whereas 'bi' means 'two', so which is more appropriate? You use the term 'bilinear' don't you? Know what a 'tetrahedron' is? So it is for biprime, triprime, tetraprime, etc.  Let me know if you really find these terms objectionable, rather than merely unconventional. There has also been some discussion of my definition of twin prime. I adopted that definition since I wanted to be unambiguous about which way you go from p, so that p a twin prime automatically means p+2 is prime. Any suggestions on how we might refer to the first and second primes in a twin prime pair? 'first-twin' and second-twin' Mmm, primo-twin? secundo-twin? How about just twin prime p and sibling prime for p+2? Email me with suggestions or correct my Latin.

D-numbers
mod 3 D-numbers
composite D-numbers

The D-numbers

Recall that a prime p is called twin if p+2 is also prime and called Sophie Germain if 2p+1 is prime. Thus, a prime p that is twin and Sophie Germain is called a Sophie Germain twin prime. The question arises, how can Sophie Germain twin primes be described in a way that encourages natural generalization? Since the set of divisors of p is {1,p}, a Sophie Germain twin prime is a prime p such that set {p+d+1| d is a divisor of p} consists of only prime numbers. Thus, we define

Definition 1. A positive integer n satisfies Property D and is called a D-number if n + d + 1 is prime for all divisors d of n. Let D denote the set of numbers that satisfy Property D.

QUESTION: Is there a better name? Please email me if you have a suggestion.

The sequence did not appear in the Online Encyclopedia of Integer Sequences so I submitted the sequence and several subsequences:

Sloane emails: (A120806)(A120807)(A120808)(A120809)(A120810)
Other Sloane submissions here.

DISCLAIMER: I retract all "structural" conjectures of the D-numbers (e.g. A120806) and now take a pragmatic approach: a D-number is where you find one.

What do D-numbers look like? They can't be even since n+2 must be prime. We need the following definitions.

Definition 1. The degree of an integer will be the sum of the exponents in its prime factorization. A biprime is an integer of degree 2, a tripime is an integer of degree 3, and a tetraprime is an integer of degree 4. [Now how about polyprime? As in 5-polyprime? Mmm? Didn't think so. ;)]

Definition 2. Two primes p and q are compatible if p<=q, say, and p does not divide q+1,otherwise incompatible. Any positive integer whose prime factors are pairwise compatible will be called compatible.

Theorem: Any D-number greater than 1 must be compatible.

The D-conjecture I. If x is a compatible odd integer not in D, then there exists an integer y such that x*y is in D.

The D-connecture II. [Draft.] Let n be a positive integer and let p1^m1*...*pk^mk be its prime factorization, and regard n as the value of the monomial f(X1,...,Xk) = X1^m1*...*Xk^mk at X1=p1,...Xk=pk. If all the polynomials f(X1,...,Xk) + d(X1,...,Xk)+1, d a divisor of f, are irreducible over Z for all possible k-1 choices among X1=p1,...,Xk=pk, then n is in D. [This is a special case of Schinzel's Hypothesis.]

Example. No number of the form 7*q^3 is in D since the polynomial 7*q^3 + q^3 + 1 = 8*q^3 + 1 factors. Generalizing, no number of the form 7*q^(3*k) is in D since 8*q^(3*k)+1 factors.

Examples of D-numbers.

Theorem:  The triprime 3*q^2 is not in D for any prime q.

Proof: Define the set S(p,q) = {p*q^2 + d + 1 | d is a divisor of p*q^2} and consider S(3,x)={3x^2+2, 3x^2+4, 3x^2+x+1, 3x^2+3x+1, 4x^2+1, 6x^2+1}. For x=5 one obtains 77, 79, 81, 91, 101, 151 and 77, 81, and 91 are not prime. Every prime other than 5 can be written as q=5k+j for some j=1,2,3,4, and it is easy to determine that S(3,j) always contains at least one element divisible by 5. Therefore, 3*q^2 is not in D for any odd prime q.

The primes p=17, 23, 47, 53, and 83 all have the property observed with 3*p^2, namely that each of the sets S(p,j), j=1,2,3,4, all contain an element divisible by 5 and the set S(p,5) has a composite so p*q^2 is not in D for any q. I call this exclusion criterion the "mod 5 test."

The text file (pq2.txt) conatins the first 50 or so primes p that fail the mod 5 test paired with their corresponding q-factors. Example: 839*(614465851)^2.

(a) p^4(5). None exist.

(b) p^3*q(8). Given a prime p, find the q such that p^3*q is in D. Here is text file for all p, 3<=p<=103, and corresponding q, (p3q.txt).

(c) p*q^3(8). Given a prime p, find the first q such that p*q^3 is in D. Search ongoing in the range 3<=p<=103. Phil Carmody has found 3*6423772250257^3. I have found a few more (8/28/06): (pq3.txt)

(d) p^2*q^2(9). If a D-number has the form p^2*q^2(9), then it must necessarily be of the form 9*q^2. Phil Carmody has found the first three primes q:

9*(88455267930709)^2

9*(94525650999739)^2

9*(377602144523101)^2

Yes, that's 88 trillion, 94 trillion, and 377 trillion! Thanks to Gerry Myerson for pointing out that, while 9 is the only prime-power square, 9*q^2 was certainly a possibility. He wrote down the 9 polynomials n+d+1 in q, and observed that, since they are irreducible over Z, by Schinzel's Hypothesis there should plausibly exist a prime q such that the 9 polynomials simultaneously have prime values.

(e) p*q*r^2(12). Given a compatible biprime p*q, find the first r such that p*q*r^2(12) is in D. Search not started.

(f) p*q*r*s(16). Given a compatible triprime p*q*r, find the first s such that p*q*r*s(16) is in D. Search not started. Currently The Holy Grail of D-numbers.

Degree 5. 5-polyprimes: p^5(6), p^4*q(10), p^3*q^2(12), p^2*q^3(12), p*q^4(10), p*q*r*s(16).

(a) p^5(6). None exist.

(b) p^4*q(10). Here are the D-numbers for p=3, 5, 7, 11, 13, 17, 19, and 23 (p4q.txt).

3^4*2714604671329

5^4*1655559005621

7^4*1704853363229

11^4*2667795560201

13^4*2327546851709

17^4*12152167742909

19^4*2213863394081

23^4*1282136158859

(c) p^3*q^2(12). Given p, find q. Search not started.

(c) p^2*q^3(12). Given p, find q. Search not started.

(c) p*q^4(10). Given p, find q. Search not started. I expect q to be in the hundred trillion range.

(c) p*q*r*s*t(32). Given p,q,r,s find t. Search not started. 

Higher degrees.

It seems that any reasonably computationally accessible D-number as of 7/31/06 has at most 10 divisors. The first example with 12 divisors is going to be astronomical. Let me know when you find one!

Mod 3 D-numbers

It is an easy exercise in modular arithmetic to show that the only prime p such that both n=p-2 and 2n+1 are divisible by p is p=3.

Definition: A positive integer n such that  n+d+1 is divisible by 3 for all divisors d of n is called a mod 3 D-number.

If you have a better idea of what to call them, just email me.

Here is a file with a list of some mod 3 D-numbers (mod3.txt). I haven't bothered to sort the file according to number of divisors, but there are some with 12, 16, and even 24 divisors!

Composite D-numbers

Definition: A positive integer n such that n+d+1 is compositie for all divisors d of n is called a composite D-number. Note that a composite D-number may be prime.

Here is a file with the first composite D-numbers less than 10^5, including primes (composite.txt). There are some with 16, 20, 24, 32, and even 36 divisors!

Let me know

Email me with anything you know about these numbers. I'll be glad to hear from you.

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Page last revised: 8/07/06.