Twin primes
The spacing between the consecutive primes follows no obvious pattern. Since 2 is the only even prime number,2 and 3 are the only pair of prime numbers with a difference 1.Since all other prime numbers except 2 are odd, the difference between any two such primes will always be even. If the Difference is 2, then such primes are called twin-primes. Up to hundred, there are 8 twin primes –
(3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61) and (71,73). The product of the twin primes with difference 2 is always one less than the square of its mean. For example,
(3,5) ; 3 x 5 = 42 -1
(5,7) ; 5 x 7 = 62 -1
If m is the mean of a twin prime, then they can be represented in terms of its mean as(m-1) and (m+1) and hence in accordance with algebra, its product satisfies m2 – 1.
Like prime numbers, twin primes are also infinite. As N increases, the number of twin primes also increases.Table.11 gives some details about it.
Table.11.Number of twin primes below N
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Up to N No.of twin primes
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100 8
1,000 35
1,000,000 8,169
1,000,000,000 3,424,506
,000,000,000,000 1,870,585,220
1,000,000,000,000,000 1,177,209,242,304
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The largest known twin primes to date are 318,032,362 x 2107,001± 1,which have 32,220 digits. Its preceding largest twin primes were 4,648,619,711,505 x 260,000± 1, which have 18,075 digits. The pair of twin primes are inexhaustible like the primes. However, there is no strict mathematical proof for it. In 1966, a Chinese mathematician, Chen-Jing gave a rough proof by assuming the two primes with a difference 2, where one is a prime and the other one is an almost prime, may be a product of two primes.
One can invent twin primes with any given difference, of course, the difference must be even. In general, the two primes p and p’ with a separation d are called pair of primes, so that
P’ = p + d
A list of pair of primes below 100 with different common difference is given in Table.12.
Table.12. Pairs of primes with different common difference
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d pair of primes No.of pairs
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4 (3,7),(7,11),(13,17),(19,23),(37,41),(43,47),(67,71)
(79,83) 8
6 (5,11),(7,13),(11,17),(13,19),(17,23),(23,29),(31,37)
(37,43),(41,47),(47,53),(61,67),(67,73),(73,79),(83,89) 14
8 (3,11),(5,13),(11,19),(23,31),(29,37),(53,61),(59,67),
(71,79),(89,97) 9
10 (3,13),(7,17),(13,23),(19,29),(31,41),(37,47),(43,53)
(61,71),(73,83),(79,89) 10
12 (5,17),(7,19),(11,23),(17,29),(19,31),(29,41),(31,43)
(41,53),(47,59),(59,71),(61,73),(67,79) 12
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It has been observed that the number of twin primes with d = 6 is nearly twice as many as twin primes with d = 2. The density of twin primes has been studied by many recreational mathematicians. It seems that twin primes become quite rare as d is made larger and larger .It opens, a new avenue in the study of prime numbers – primes in arithmetic progression.
Primes in arithmetic progression
The idea of pair of primes can be extended to get a sequence of primes with a given common difference. Primes can form arbitrarily long and short arithmetic progression. Short arithmetic progression of primes is easy to find out. For example, the primes 3, 5, 7 form a progression of 3 terms with a common difference of 2, where as 3,7,11 with a common difference of 4 and 5, 11, 17, 23, 29 with a common difference of 6 and so on.
The primes 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879 and 2089 constitute a progression of 10 terms with a common difference of 210. For longer progression, the final prime and the common difference escalate rapidly and making them difficult to find. In 1983, an amateur mathematician Paul Pritchard at Cornell , USA 4,180,566,390. In this sequence of primes, they are not consecutive. Some mathematicians have even conjectured that there are arbitrarily long arithmetic progressions of consecutive primes. For example, the consecutive primes 1741, 1747, 1753, and 1759 for a four term progression with a common difference of 6.
One can find palindromic primes in arithmetic series. A palindromic number is one which reads same, irrespective of whether we begin from the right of from the left. 99,818,2442,23532,123321...........are few examples for palindromic numbers. The palindromic numbers can have any digital root and they can be either even or odd. Palindromic numbers may even be prime. There is no two digits palindromic numbers except 11. Among the three digits numbers, there are 14 palindromic primes. They are 101,131,151,181,191,313,353,373,383,727,757,787,919,929. Palindromic primes may also form a progression, for example 10301,13331,16361,19391 and 70607,73637,76667,79697 are two typical palindromic prime series with a common difference of 3030.
The pairs of numbers n and n+6 are called sexy primes as such pairs are more likely to exist as couples in the society. In fact, the word ‘sexy’ comes from the Latin for six. As the couples reproduce their generation, the sexy pairs also sometimes do, a small family has 3 members, called sexy triplets, while the family with four members is called sexy quadruplet. For example, the sequences of sexy triplets are
17-23-29
31-37-43
47-53-59
Sexy quadruplets are,
5-11-17-23
11-17-23-29
41-47-53-59
61-67-73-79
It is observed that the first quadruplet in the above list is the only quadruplet that does not start with a number ending with 1. The sexy quintuplet 5-11-17-23-29 is unique, where one of the numbers ends with 5. There is ironically, no sexy sextuplet.
Magic squares with prime numbers
An unlimited amusement lies in getting such set of primes in arithmetic progression and incorporating them in magic squares of any given order. The magic square is a square matrix of numbers where the sum of the numbers in all the rows, columns and the main diagonals remain same. 2x2 magic square with or without prime numbers is not possible. But the 3 x 3 magic square and higher orders are possible. To construct a 3 x 3 magic square all with prime numbers, we need 9 primes in an arithmetic progression. Using the series of prime numbers starting with 199 and with a common difference 210, we can construct the following 3 x 3 magic squares with prime numbers only.
____________________ ___ ______________________
1879 1039 199 2089 1249 409
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829 619 1669 1039 829 1879
_______________________ ______________________
(a) (b)
magic constant – 3117 magic constant – 3747
Fig.3
Since it is very difficult to get a set of 9 primes in arithmetic series to construct 3 x 3 or higher order magic squares with more numbers in the series. We can think of broken series, each comprising 3 primes, but all of them with same common difference “d”, particularly in the 3 x 3 magic square. Let (a,a+d,a+2d), (b,b+d,b+2d) and (c,c+d,c+2d) be the three set of primes in arithmetic series. The magic square with them is shown in Fig.4.a. It is noted that the magic constant is a+b+c+3d for all
a + d | c | b+2d |
c+2d | b+d | a |
b | a+2d | c+d |
269 | 443 | 41 |
23 | 251 | 479 |
461 | 59 | 233 |
(a) (b) Fig.4
rows, columns and one of the diagonals. In the other diagonal it is 3(b+d). To make this sum to be equal to the magic constant, the required condition is a+d = 2b, which means that a,b,c must be in an arithmetic series, not necessarily with the same common difference by which the individual series is made up of. One typical example is (23,41,59), (233,251,269), (443,461,479),where the common difference of each series is 18 and the common difference between a,b,c is 210. In all the 3 x 3 magic squares, the magic constant is 3 times the central number. In the typical example shown above, it is 3 x 251 = 753. With this we can show that, the magic constant can be expressed as a sum of three primes in 8 different ways with the primes used in the construction of the 3 x 3 magic square.
753 = 269 + 443 + 41 = 23 + 251 + 479 = 461 + 59 + 233
= 269 + 23 + 461 = 443 + 251 + 59 = 41 + 479 + 233
= 269+ 251 + 233 = 41 + 251 + 461
One of the significant properties of magic square is the sum of squares of the numbers in extreme rows and extreme columns are same. In Fig.4.b, we have,
2692 + 4432 + 412 = 4612 + 592 + 2332
2692 + 232 + 4612 = 412 + 4792 + 2332
Few more examples for 3 x 3 magic squares with prime numbers only are given in Fig.5.
7 61 43 59 53 101 193 199 61
73 37 1 113 71 29 19 151 283
31 31 67 41 89 83 241 103 109
(a) (b) (c)
Magic constant:
37x3 = 111 71x3 = 213 151x3 = 453
Fig.5
Remember, one cannot make the magic constant to be a prime in 3 x 3 magic square filled with primes only, because it will be 3 times the central prime.
Of course, one can construct any higher order magic square with all numbers associated with it are prime. In this annotation it is exemplified with 4 x 4 magic squares. The four broken series with Four members in each, have the general form x, x + D, x+ 2D, x + 3D, where D is the common difference and x = a,b,c or d. The distribution of these numbers in the 4 x 4 magic square is illustrated in Fig.6.a along with a specific example.
a + 2D | D | C + 2D | B + 3D | 41 | 59 | 97 | 271 | |
b + 2D | c + 3D | d + D | a | 241 | 127 | 89 | 11 | |
d + 3D | a + 2D | b | c + D | 149 | 71 | 181 | 67 | |
c | b + D | a + 3D | d + 2D | 37 | 211 | 101 | 119 |
(a) Fig.6 (b)
The magic constant of the 4 x 4 magic square shown in Fig.6(a) is a +b+c+d+6D and it is same for all rows, columns and main diagonals. Since there is no condition, we can replace any one of the series by a different series starting with a different number but with same common difference. For example the series (349,379,409,439), (881,911,941,971), (907,937,967,997) are all in arithmetic progression with the same common difference 30. When you replace a series with an another series, the magic constant will be different.
Magicness of 4 x 4 magic square
It is found that the sum of the numbers in all the columns, rows and main diagonals of a 4 x 4 magic square is same. In Fig.6 (b), it is 468.
With numbers in rows with numbers in columns with numbers in diagonals
41 + 59 + 97 + 271 = 468 41 + 241 + 149 + 37 = 468 41 + 127 + 181 + 119 = 468
241 + 127 + 89 + 11 = 468 59 + 127 + 71 + 211 = 468 271 + 89 + 71 + 37 = 468
149 + 71 + 181 + 67 = 468 97 + 89 + 181 + 101 = 468
37 + 211 + 101 + 119 = 468 271 + 11 + 67 + 119 = 468
The same magic constant is exhibited by the four corner numbers in 2 x 2 squares,
41 + 59 + 241 + 127 = 468
97 + 271+ 89 + 11 = 468
149 + 71 + 37 + 211 = 468
181 + 67 + 101 + 119 = 468
Quite amazingly, the same magic constant is predicted by many set of numbers. The numbers in broken diagonals,
241 + 59 + 101 + 67 = 468
149 + 211 + 97 + 11 = 468
The numbers in four corners,
41 + 271 + 37 + 119 = 468
The numbers in the central 2 x 2 square,
127 + 89 + 71 + 181 = 468
Extreme numbers in the inner rows and inner columns,
241 + 149 + 11 + 67 = 468
59 + 97 + 211 + 101 = 468
It is found that the sums of the squares of the numbers in the two extreme rows, two extreme columns, two inner rows and two inner columns are same.
Extreme rows : 412 + 592 + 972 + 2712 = 372 + 2112 + 1012 + 1192
Inner rows : 2412 + 1272 + 892 + 112 = 1492+ 712 + 1812 + 672
Extreme columns : 412 + 2412 + 1492 + 372 = 2712 + 111 + 672 + 1192
Inner columns : 592 + 1272 + 712 + 2112 = 972 + 892 + 1812 + 1012
Prime desert
Suppose we want to find a sequence of say 35 numbers in a row with no primes among them. Consider the sequence of numbers
36! + 2, 36! + 3, 36! + 4 ...... 36! + 36
In this series of consecutive numbers, none of them will be prime. This can be mathematically understood. 36! is the product of all numbers from 1 to 36 and hence it is divisible by all of them. From this knowledge we can say that 36! + 2 will be divisible by 2, 36! + 3 will be divisible by 3 ......... 36! + 36 will be divisible by 36. Note that we cannot extend this in either side. Hence 36! + 1 might be a prime.
If we want 1000 numbers in a row without a prime, this can be achieved by taking 1001! + 2, 1001! + 3,.......... 1001! + 1001. A set of such long consecutive numbers devoid of prime numbers is called prime desert.
Consecutive prime pairs
Consecutive prime pairs are two or more successive twin primes with a particular common difference d. That is there should not be any prime in between any two such pairs. For example,
(101,103), (107,109) with d = 2
(307,311), (313,317) with d = 4
(23,29), (31,37) with d =6
With the help of a computer it is easy to prepare a list of prime pairs (p, p+d) where p is a prime and d an even integer. When n tends to infinity, it s found that n(d)/n(2) , the ratio of the number of prime pairs with common difference d to the number of prime pairs with common difference 2 is nearly equal to unity for d = 4,8,16,32,64,..... in the form of some power of 2. But when d is a multiple of 6 like 6, 12, 18, 24, 36, 48, 54,..... n(d)/n(2) is nearly equal to 2.
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