Some families of solutions: Suppose n satisfies the property. And suppose that for some p, p+1 divides n+1, while for all prime factors q | n, (q+1) | (p+1)*(p-1). Then n*p*p also satisfies the property: n*p*p + 1 = (n+1)*p*p - (p+1)*(p-1) Example: n = 7 trivially satisfies the property. p = 3 satisfies the above condition. So 7*3*3 also satisfies the property. Now for n = 7*3*3, both 3 and 7 satisfy the above condition, so by repetition, 7^(2j+1) * 3^(2k) satisfies the property. Another example: n = 3*7*19, p = 19.
There exists a composite integer n such that for each prime divisor p of n (p+1)|(n+1).
If it is true then what is the smallest such number? Are such numbers are infinitely many?
I would be extremely grateful for your help.
With best regards,
Nayandeep Deka Baruah Dept. of Math. Sciences, Tezpur University Napaam-784028 Assam, INDIA.
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