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Answer : I believe your sequence continues forever but grows quickly. If n n n is large, the density of primes around n n n is log ⁡ n \log n lo g n . Since log ⁡ n \log n lo g n is so much smaller than n n n , the chance a random n n n has k k k arrows is about 1 ( log ⁡ n ) k + 1 \frac 1{(\log n)^{k+1}} ( l o g n ) k + 1 1 ​ . The expected number of sequences of length k k k above 1 0 12 , 10^{12}, 1 0 12 , say, is then ∫ 1 0 12 ∞ d n ( log ⁡ n ) k + 1 \int_{10^{12}}^\infty \frac {dn}{(\log n)^{k+1}} ∫ 1 0 12 ∞ ​ ( l o g n ) k + 1 d n ​ . This diverges because ( log ⁡ n ) k (\log n)^k ( lo g n ) k becomes less than n n n for n n n large enough and we know the integral of 1 n \frac 1n n 1 ​ diverges. Each subtraction is only of order n log ⁡ n \frac n{\log n} l o g n n ​ , which is small compared to n n n and the log will not change much. If we ask what length of sequence we expect to find among the 12 12 12 digit numbers, we note that the log of these numbers ...

Answer : The documentation of integral_points cites the algorithm found in Henri Cohen, Number Theory, Vol. I: Tools and Diophantine Equations. GTM 239, Springer, 2007. You can read the implementation e.g. by entering E.integral_points?? into a SageMath session. The algorithm finds all the integral points, but it depends crucially on having computed the Mordell-Weil group. On this topic, the documentation reads: mw_base - list of EllipticCurvePoint generating the Mordell-Weil group of E (default: ‘auto’ - calls self.gens() ) Note: The complexity increases exponentially in the rank of curve E. The computation time (but not the output!) depends on the Mordell-Weil basis. If mw_base is given but is not a basis for the Mordell-Weil group (modulo torsion), integral points which are not in the subgroup generated by the given points will almost certainly not be listed. In turn, the documentation of gens() , about its optional argument proof , says: p...