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

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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