Quandaries and Queries


Who is asking: Student
Level: Secondary

at least it seems like for any integers n and k,
10k+3 and 10k+7 do not divide n²+n-1
I tested this for every n from 0 to 3200 (which means same for the numbers from -3201 to -1)
could this be true, or is it just coincidence, or am I just totally wrong?



Hi Arne,

This is a nice observation. It is in fact true, the units digit of a divisor of n² + n - 1 can not be 3 or 7. The proof I have shows that the result is true for prime factors and then uses properties of a particular quadratic form to exend he result to all factors.

Thanks for the question,