Numbers: Their Tales, Types, and Treasures.

Chapter 8: Special Numbers

8.3.PLAYING WITH PRIMES

There are some patterns among prime numbers. For example, there are reversible prime numbers—that is, those prime numbers whose reverse is also a prime number: 13 and 31, 17 and 71, 37 and 73, 79 and 97, 107 and 701, 113 and 311, 149 and 941, 157 and 751.

Referring back to our previous discussion of palindromic numbers, there are also palindromic numbers that are prime numbers, such as: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, and 13331.

Moreover, there are repunit numbers that are also prime, such as 11, 1111111111111111111, and 11111111111111111111111, with the next two such prime repunit numbers having large numbers of 1s, specifically, 317 and 1,031 1s.

There are prime numbers that have the characteristic that any other arrangement of the digits will also produce a prime number. The first few of these are: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, and 991. It is believed that larger such primes are repunit primes.

There are also prime numbers that remain prime numbers even when their digits are moved in a circular fashion. For example, the prime number 1,193 can have its digits “rotated” to form the following numbers: 1,931; 9,311; 3,119. Since all of these rotated-digit variations yield a prime number, we call the number 1,193 a circular prime number. Other such circular prime numbers are: 2; 3; 5; 7; 11; 13; 17; 31; 37; 71; 73; 79; 97; 113; 131; 197; 199; 311; 337; 373; 719; 733; 919; 971; 991; 1,193; 1,931; 3,119; 3,779; 7,793; 7,937; 9,311; 9,377; 11,939; 19,391; 19,937; 37,199; 39,119; 71,993; 91,193; 93,719; 93,911; and 99,371.

Continuing this exploration of numerical oddities, we arrive at the amazing prime number 193,939. This is also a circular prime number because all of its rotated-digits numbers are prime numbers:

193,939; 939,391; 393,919; 939,193; 391,939; and 919,393.

But in this case there is another arrangement of these digits that results in another circular prime number:

199,933; 999,331; 993,319; 933,199; 331,999; and 319,993.

On the other hand, there are some prime numbers that, when any one of their digits is changed to another value, will always result in a composite (nonprime) number. Some of these are:

294,001; 505,447; 584,141; 604,171; 971,767; 1,062,599;
1,282,529; 1,524,181; 2,017,963; 2,474,431; 2,690,201;
3,085,553; 3,326,489; and 4,393,139.

There are also prime numbers that are the sum of two consecutive squares—that is, n² + (n + 1)². The first few of these so-called sum-of-consecutive-squares primes are: 1 + 4 = 5, 4 + 9 = 13, and 16 + 25 = 41, and the rest of the first few are: 61; 113; 181; 313; 421; 613; 761; 1,013; 1,201; 1,301; 1,741; 1,861; 2,113; 2,381; 2,521; 3,121; 3,613; 4,513; 5,101; 7,321; 8,581; 9,661; 9,941; 10,531; 12,641; 13,613; 14,281; 14,621; 15,313; 16,381; 19,013; 19,801; 20,201; 21,013; 21,841; 23,981; 24,421; and 26,681.

Then there are additive primes, which are prime numbers where the sum of their digits is also a prime number. Some of these are: 2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 113, and 131.

In recreational mathematics we often seek strange characteristics of numbers for entertainment. For example, there is a set of prime numbers that is called minimal primes. It is believed that there are only twenty-six such prime numbers, with the characteristic that any subsequence of the digits will be a composite number (a nonprime number). To better understand what is meant by “minimal primes,” let us consider one example. For the prime number 6,949 we can establish the following numbers, taking the digits in sequence as follows: 6, 9, 4, 69, 94, 49, 64, 99, 694, 699, and 949, all of which are composite numbers. The list of the twenty-six known minimal primes is as follows: 2; 3; 5; 7; 11; 19; 41; 61; 89; 409; 449; 499; 881; 991; 6,469; 6,949; 9,001; 9,049; 9,649; 9,949; 60,649; 666,649; 946,669; 60,000,049; 66,000,049; and 66,600,049.