Find all natural values of n which make the value of (n^3 - 1)/5 a prime number.
Find all values of p which make p, p + 2 and p + 4 prime numbers.
Find all prime numbers p which also make p + 10 and p + 20 prime too.
Thanks in advance!
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1Problem two HINT: Consider modulo 3. – K.defaoite Dec 11 '20 at 00:43
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2ditto problem three – user2661923 Dec 11 '20 at 01:21
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Hint on (1): $x=\frac{n^3-1}{5}=\frac{(n-1)(n^2+n+1)}{5}$. $x$ (if it is an integer) has integer factors $>1$ and is not prime unless either $n-1=1$ or $\frac{n-1}{5}=1$ – Keith Backman Dec 11 '20 at 16:41
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$\hspace0.5cm$ (1) It is rather easy to show that the only $n$ is $6$.
$\hspace0.5cm$ (2) A prime $p\gt3$ is of the form $3n+1$ or $3n+2$ so we have $$3n+1,3n+3,3n+5 \text { or }3n+2,3n+5,3n+6$$ In both cases one of the three numbers is divisible by $3$. Thus the only example is $(p,p+2,p+4)=(3,5,7)$.
$\hspace0.5cm$ (3) Proceding as in (2) we have $$3n+11,3n+21\text { or }3n+12,3n+22$$ and again one of the numbers will be multiple of $3$.
Piquito
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