## When (m,n) and (m+1,n+1) have the same prime divisors

Find pairs of different positive integers m and n such that

>>   $(m, \; n)$   have the same prime divisors
>>   $(m+1, \; n+1)$   have the same prime divisors

For example,

$(m, \; n) = (75, \; 1215)$ ;        $(m+1, \; n+1) = (76, \; 1216)$

$75 = 3\times 5^2$ ………………. $1215 = 5\times 3^5$
$76 = 2^2\times 19$ ……………… $1216 = 2^6\times 19$

Other examples include:

$2$   is a prime number …………   $8 = 2^3$
$3$   is a prime number …………   $9 = 3^2$

$6 = 2\times 3$   ………………………….   $48 = 2^4\times 3$
$7$   is a prime number …………..   $49 = 7^2$

$14 = 2\times 7$   …………………   $224 = 2^5\times 7$
$15 = 3\times 5$   …………………   $225 = 3^2\times 5^2$

$30 = 2\times 3\times 5$   …………………….   $960 = 2^6\times 3\times 5$
$31$   is a prime number …………..   $961 = 31^2$

$62 = 2\times 31$   ………………….   $3968 = 2^7\times 31$
$63 = 3^2\times 7$   ………………….   $3969 = 3^4\times 7^2$

$126 = 2\times 3^2\times 7$   ………………..   $16128 = 2^8\times 3^2\times 7$
$127$   is a prime number………..   $16129 = 127^2$

$254 = 2\times 127$   ………………..   $65024 = 2^9\times 127$
$255 = 3\times 5\times 17$   ……………   $65025 = 3^2\times 5^2\times 17^2$

$510 = 2\times 3\times 5\times 17$   ………..   $261120 = 2^{10}\times 3\times 5\times 17$
$511 = 7\times 73$   ……………………   $261121 = 7^2\times 73^2$

$1022 = 2\times 7\times 73$   …………..   $1046528 = 2^{11}\times 7\times 73$
$1023 = 3\times 11\times 31$   ………….   $1046529 = 3^2\times 11^2\times 31^2$

$2046 = 2\times 3\times 11\times 31$   ………   $4190208 = 2^{12}\times 3\times 11\times 31$
$2047 = 23\times 89$   ………………….   $4190209 = 23^2\times 89^2$

$4094 = 2\times 23\times 89$   ………….   $16769024 = 2^{13}\times 23\times 89$
$4095 = 3^2\times 5\times 7\times 13$   ……..   $16769025 = 3^4\times 5^2\times 7^2\times 13^2$

$8190 = 2\times 3^2\times 5\times 7\times 13$   ….   $67092480 = 2^{14}\times 3^2\times 5\times 7\times 13$
$8191$   is a prime number…………   $67092481 = 8191^2$

$16382 = 2\times 8191$   ………………   $268402688 = 2^{15}\times 8191$
$16383 = 3\times 43\times 127$   ………..   $268402689 = 3^2\times 43^2\times 127^2$

$32766 = 2\times 3\times 43\times 127$   …….   $1073676288 = 2^{16}\times 3\times 43\times 127$
$32767 = 7\times 31\times 151$   …………..   $1073676289 = 7^2\times 31^2\times 151^2$