Number 2 – an indifferent prime
Number 2 has many odd characteristics among all prime numbers. It is the only even prime and cannot be classified under any group. It does not fit with the general expression representing a prime number as the difference of two squares. But it can be expressed as the difference of two squares of fractional numbers,
2 = (1.5)2 - (0.5)2
In order to express a number as the difference of two squares, one should be able to express the number considered as the product of two factors having same parity, that is both must be odd or both must be even. 22 cannot be accommodated in this description, because its factors 1 and 4 do not have same parity. But 23 , with factors 2 and 4 suits well.
23 = 4 x 2 = [(4+2)/2]2 - [(4-2)/2]2 = 32 - 12
Similarly one can describe the higher powers of 2
24 = 16 = 2 x 8 = 52 - 32 ; 27 = 128 = 2 x 64 = 332 - 312
= 4 x 4 = 42 - 02 ; = 4 x 32 = 182 - 142
25 = 32 = 2 x 16 = 92 - 72 ; = 8 x 16 = 122 - 42
= 4 x 8 = 62 - 22 ; 28 = 256 = 2 x 128 = 652 - 632
26 = 64 = 2 x 2 = 172 -152 ; = 4 x 64 = 342 - 302
= 4 x 16 = 102 - 62 ; = 8 x 32 = 202 - 122
= 8 x 8 = 82 - 02
We can extend this idea to any required power of 2 and other prime numbers also. For example in the case of 17,
17 = 1 x 17 = 92 - 82
172 = 289 = 1 x 289 = 1452 - 1442
173 = 4913 = 1 x 4913 = 24572 - 24562
= 17 x 289 = 1532 - 1362
Number of ways (N) by which the power (m) of a prime number (p) can be expressed in this fashion is given by (m+1)/1 for odd m and m/2 for even m. In the case of 2, 2 and 22cannot be expressed like this and hence for 2, m must be reduced by 2.
From this it is clear that not only the prime number, but also its square can be expressed as the difference of two squares in only one way. Further, this idea can be applied even to the product of any two or more distinct primes. If n distinct primes are multiplied together, then the total number of ways by which it can be expressed as the product of two numbers will be 2n-1 and consequently it can be represented as the difference of two squares by the same number of ways. 2 is the only prime, twice of which is a square, i.e.,
2 + 2 = 22
No other primes have this property. If we suppose that P has such property, then,
2 P = P’2 = P’ x P’
Since P’ and P have no common factor, P must be divisible by P’, which is against the primality of P.
However, 2 combining with many other primes give odd square numbers. For example,
2 + 7 = 9 = 32 ; 2 + 223 = 225 = 152
2 + 23 = 25 = 52 ; 2 + 359 = 361 = 192
2 + 47 = 49 = 72 ; 2 + 439 = 441 = 212
2 + 79 = 8l = 92 ; 2 + 727 = 729 = 272
2 + 167 = 169 = 132 ; 2 + 839 = 841 = 292
Not only two, but also other primes combining with different primes give square numbers. But they are all even in nature. For example,
3 + 193 = 196 = 142 ; 3 + 1021 = 1024 = 322
3 + 397 = 400 = 202 ; 3 + 1153 = 1156 = 342
3 + 673 = 676 = 262 ; 3 + 1597 = 1600 = 402
Any two odd primes are added together, it results with an even number and hence the resulting square and its root are also even in nature. That is, representation of odd squares by the summation of two primes cannot be made without the even prime 2 and only even squares can be represented with primes excluding 2.
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