SOME UNRESOLVED PROPERTIES OF PRIME NUMBERS
The field of prime numbers is so vast and its content is ever increasing with unresolved properties of them. Some of them are illustrated here.
1. Every positive integer can be expressed as a product of primes .The factorisation is unique except for a rearrangement of the factors. If the positive integer assumed is prime, it can be shown as a product of the prime and 1. (If number 1 is not taken in the list of prime numbers, then all the prime numbers cannot be expressed as the product of primes as shown for composite numbers).Few examples are given below
33 = 3 x 11
63 = 3 x 37
132 = 2 x 2 x 3 x 11
210 = 2 x 3 x 5 x 7
222 = 2 x 3 x 37
265 = 5 x 53
Christian Goldbach, the Prussian mathematician pointed out that every even number greater than 2 is a sum of two primes.
4 = 1 + 3 18 = 7 + 11
6 = 1 + 5 20 = 7 + 13
8 = 1 + 7 22 = 5 + 17
10 = 3 + 7 24 = 5 + 19
12 = 5 + 7 26 = 3 + 23
14 = 7 + 7 28 = 5 + 23
16 = 5 + 11 30 = 1 + 29
Number theorists have verified Goldbach’s conjecture for all even numbers up to 108, but without proof. However, it paves a way to know other unresolved properties of primes. Every odd integer greater than 7 can be shown as a sum of three odd primes, by many ways.
11 = 1+3+7 = 3+3+5 = 1+5+5
13 = 3+3+7 = 1+5+ 7 = 3+5+ 5
17 = 3+7+7 = 1+5+11 = 3 +7 +7
19 = 5+7+7 = 3+5+11 = 3+3+13
2. Most of the prime numbers can be expressed as a sum of a prime and twice a square number. e.g.,
11 = 3 + 2 x 22 = 3 + 23
13 = 5 + 2 x 22 = 5 + 23
17 = ............. . = 1 + 24
19 = 11+ 2 x 22 = 3 + 24
23 = 5+ 2 x 32 = 7 + 24
29 = 11+ 2 x 32 = 13 + 24
31 = 13+ 2 x 32 = 23 + 23
37 = 19+ 2 x 32 = 5 + 25
3. There exists at least one prime number between consecutive triangular or square numbers. Triangular numbers are the numbers obtained by summing up all the numbers in natural series up to a number. First few triangular numbers are 1,3,6,10,15,21,28,36,45,55,66,78,91,105,120........ Similarly the first few square numbers are 1,4,9,16,25,36,49,64,81,100,...... One can check and confirm the captivity of prime or primes between any two successive triangular or square numbers.
4. We have already come across the Euclidian concept on infinite prime numbers. Even though it is unique and rigid, it is not possible to find all the primes with the help of this method.
The resultant of the product of successive primes is even by the presence of 2. The oddness of the generated prime is provided by the subsequent addition of 1. In contrary, if this 1 is subtracted from the product of primes, it will not generate higher prime. For example,
2 x 3 x 5 x 7 – 1 = 209 = 19 x 11
One can add or delete one or more primes in the product of primes, even then higher prime numbers are generated on addition of 1, but not always (remember the presence of 2 in the product is necessary, it cannot be deleted ). It is illustrated below:
Valid delete : 2 x 3 x 7 + 1 = 43 (5 is deleted)
2 x 5 x 7 + 1 = 71 (3 is deleted)
2 x 3 x 17 + 1 = 103 ( 17 is added)
2 x 3 x 43 + 1 =1807 ( 43 is added)
Invalid delete: 2 x 3 x 19 + 1 = 115 = 5 x 23
2 x 3 x 29 + 1 = 175 = 5 x 5 x7
2 x 3 x 13 x 5 + 1 = 391 = 17 x 23
2 x 3 x 17 x 5 + 1 = 511 = 7 x 73
5. Ramanujan discovered that the product of consecutive primes starting from 2 up to some prime numbers starting from 2 up to some prime numbers, when added with ¼ gives a square of a mixed fraction, the fractional part being ½. For example,
2 + ¼ = 2 ¼ = (1 ½)2 = 1 x 2 + ¼
2 x 3 + ¼ = 6 ¼ = (2 ½ )2 = 2 x 3 + ¼
2 x 3 x 5 + ¼ = 30 ¼ = (5 ½ )2 = 5 x 6 + ¼
2 x 3 x 5 x 7 + ¼ = 210 ¼ = (14 ½ )2 = 14 x 15 + ¼
2 x 3 x 5 x 7 x 11 + ¼ = 2310 ¼ = ......... = ............
2 x 3 x 5 x 7 x 11 x 13 + ¼= 30030 ¼ = ........ = ............
2310 ¼ and 30030 ¼ cannot be expressed as squares of a mixed fraction. This is because, the numbers 2310 and 30030 cannot be expressed as the product of two successive numbers in the natural series. This can be clearly understood with the following delightful pattern among the square of mixed fractions.
1.5 x 1.5 = 2.25 = 1 x 2 + 0.25
2.5 x 2.5 = 6.25 = 2 x 3 + 0.25
3.5 x 3.5 = 12.25 = 3 x 4 + 0.25
4.5 x 4.5 = 20.25 = 4 x 5 + 0.25
5.5 x 5.5 = 30.25 = 5 x 6 + 0.25
6.5 x 6.5 = 42.25 = 6 x 7 + 0.25
The divisors of 2310 are 1, 2 ,3, 5, 7 and 11. Hence 2310 can be expressed as the product of 42 x 55 or 77 x 30, but cannot be expressed as the product of any two successive numbers in the natural series, which shows that 2310 ¼ cannot be represented as a square of a mixed fraction ,the fractional part being ½ .
2x3x5x7x11x13x17 = 510510
510510 can be expressed as the product of two consecutive numbers,
510510 = 714 x 715
Hence
510510 ½ = ( 714 ½ )2
But 2x3x5x7x11x13x17x19= 9,699,690, where 9699690 ¼ cannot be expressed as a square of mixed fraction.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49 ......
The distribution of prime numbers in the diagonal of square spiral can be expressed in a quadratic form, which resembles the polynomials. For example, when we start from 1, the sequence of prime numbers 7,23,47,79..... that fall in one of the descending diagonals, can be obtained by substituting values of x in the relation 4x2 + 4x – 1.The Spiral that starts from 41 reveals Euler’s formula
x2 + x +41.
105 104 103 102 101 100 99 98 97
106 77 76 75 74 73 72 71 96
107 78 57 56 55 54 53 70 95
108 79 58 45 44 43 52 69 94
109 80 59 46 41 42 51 68 93
110 81 60 47 48 49 50 67 92
111 82 61 62 63 64 65 66 91
112 83 84 85 86 87 88 89 90
113 114 115 116 117 118 119 120 121
The prime numbers, 421,347,281,223,173,131 (not shown in Fig.2) and 97, 71, 53, 43, 41, 47, 61, 83, 113 (shown in Fig .2), 151,197,251,313,383 (not shown) that fall in the ascending main diagonal can be obtained by Euler’s formula.
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