This is valid till the prime number 5 is sufficient to test the powerfreeness of another integer. {m,n,k} are numbers/integers in the vicinity and for very high powers in Fermat’s theorem one needs higher prime numbers. This theorem has been evidently tested for small powers.

# powerfree numbers..

## new inventions in number theory .. [summary]

1. a number is not a power of 6 if it’s last two digits are not one of these: 16, 36, 56, 76 or 96, always … OR integer whose 2nd digit to left from right is any odd numbers less than 10 (1,3,5,7..etc)

2. a number is not a power of 5 if it’s last two digits are not 25

3. for powers of 4, always either a 4 or a 6 in the last digit (which adds upto 10)

4. powers of 7: last digit is always an odd number less than 10 but not 5.

5. for powers of 3 is same as that of 7 but their 2nd last digit differ..

## Interesting observations in number theory

(perhaps never known before at-least most of these) I will post another article for my 50cent theorem and related facts an interesting pattern for powers of 6: (6^2 .. 6^3.. 6^12), the last two digits you get by subtracting 20, for all powers from 2 to 12 and more The last two digits I have separated by hyphen – ; 36, 2-16, 12-96, 77-76, 466-56, 2799-36, 16796-16, 100776-96, 604661-76, ***56, ***36, ***16, ***96 …. Therefore you can discard all numbers whose last 2 digits are not 16, 36, 56, 76 or 96 : I call this my 50 cent axiom for power exponents of 6.. note that 20+16 =36, so start with 16 and go on adding 20.. new 50cent axiom for powers of 6, a number is not a power of 6 if it’s last two digits are not one of these: 16, 36, 56, 76 or 96, always… for 6 one can also note it’s all odd numbers