De Polignac Numbers

Today I turned 25691 days old and one of the properties of this number is that it's a de Polignac number. Moreover, it's the smaller of a pair of such numbers with the larger member of the pair being 25693. Numbers Aplenty must hold these numbers in high regard because, for every odd number searched for, it makes a point of declaring whether it is de Polignac or not. Figure 1 and Figure 2 illustrate this.

Figure 1: screenshot from source showing that
25689 is not a de Polignac number

Figure 2: screenshot from source showing that
25691 is a de Polignac number

Numbers Aplenty defines a de Polignac number as follows:

A de Polignac number is an odd number \(n\) that cannot be expressed as \(n=2^k+p\), for \(p\) prime.

This definition is equivalent to there being no prime \(p=n-2^k\) for values of \(2^k<n\). In the case of \(25689\), it is shown that \(2^4\) when subtracted from \(25689\) produces the prime \(25673\). Therefore \(25689\) is not a de Polignac number. In the case of \(25691\), it looks as if the expression \(2^k-25691\) should be reversed and instead read \(25691-2^k\).

Some interesting facts about de Polignac numbers include:

• There are many de Polignac consecutive odd numbers that differ by 2. The earliest ones are: 905 & 907. Others are 3341 & 3343; 3431 &3433; and so on. However, there are no de Polignac twin primes. Source.
• In the interval 5 to 50,000,000 there are about 21.8% (477445/2188591) of de Polignac numbers that are prime and this number slowly decreases. Thus composite de Polignac are more abundant than prime de Polignac. Source.
• Paul Erdös proved that there are an infinity of de Polignac numbers, for example all the numbers of the form 1260327937 + 2863311360⋅\(k\). Source.
• The smallest composite de Polignac number is 905, while the first square is 40401. Source.
• The smallest 3 × 3 magic square whose entries are de Polignac numbers is shown in Figure 3. Source.

Figure 3: magic square of de Polignac numbers

•  \(10^{999} +18919\) is the earliest titanic Polignac prime number. The next are:

\(10^{999} + 25561\)

\(10^{999} + 28047\)

\(10^{999} + 28437\)

\(10^{999} + 41037\)

\(10^{999} + 55587\)

\(10^{999} + 63177\)

\(10^{999} + 73209\)

\(10^{999} + 75301\)

\(10^{999} + 90079\)
Source

• \(2^{2^n} - 5\) is a de Polignac number for each n > 2 e.g. 251, 65531, 4294967291, ... Source.

Here are some biographical details concerning the mathematician after whom these numbers are named. Source.

Alphonse de Polignac (1826–1863) was a French mathematician. In 1849, the year he was admitted to Polytechnique, he made what's known as Polignac's conjecture: 

For every positive integer \(k\), there are infinitely many prime gaps of size \(2k\). The case \(k = 1\) is the twin prime conjecture. 

His father, Jules de Polignac (1780-1847) was prime minister of Charles X until the Bourbon dynasty was overthrown (1830).

Here a permalink to SageMathCell and below is the SageMath code that I developed to identify de Polignac numbers within a given range (in the example, between 25500 and 25600):

INPUT 

upper=26000
power=ceil(log(upper, 2))
NP=[]
for p in prime_range(2, upper):
    for k in [0..power]:
        if (2^k+p)<upper:
            NP.append(2^k+p)
N=[x for x in [1..upper]]
for x in (Set(N)-Set(NP)):
    if x%2==1:
        if x>25600 and x<26000:
            print x, 

OUTPUT 

25627 25631 25691 25693 25723 25739 25747 25753 25777 25783 25799 25829 25841 25909 25925 25961 25979 25987 25993