This number 15^6 has a long sequence of positive consecutive integers. The last term is a 4-digit prime number whose first digit(from the left) x second digit x third digit - fourth digit = 101. What are the first term, last term and the number of terms of this sequence? Thank you for help.

Guest Aug 22, 2019

#1**0 **

Pls check your problem for typos, because 15^{6} = 11,390,625.

Not only is 0625 not prime, but all the other conditions fail too.

Maybe I'm misunderstanding the problem? Unlikely.

.

Guest Aug 22, 2019

#2**0 **

The sequence doesn't start at 1 and end at 15^6. It starts somewhere else and ends in a 4-digit integer, which is a prime number. You didn't understand the question.

Guest Aug 22, 2019

#3**+1 **

Since there are over 1000 prime numbers between 1,000 and 10,000, I know of no formula to narrow this down to one or two numbers except by a short computer code that will eliminate 99% of them. I wrote a short code that narrowed them down to just 3: 6,367, 6,637, 9,437. By a simple test using this modified AP formula, it turn out that the last term is: 6,637.

** N/2 * ((4*15^6 / N) - (2*6637) + (1*(N-1)) = 15^6, Solve for N Plugging it in the above formula and using the quadratic formula, it gives N=2,025 which is the number of terms; 6,637 - 2,025 + 1 =4,613 which is the first term. [4,613 + 6,637] / 2 * 2,025 =15^6 which checks out.**

Guest Aug 22, 2019