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**1 - 4**of**4**### A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO |x 3 − y 2 | VALUES

"... Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = |x 3 −y 2 | values. Seventeen new values of k

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Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = |x 3 −y 2 | values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in other cases, it seems difficult to achieve small absolute values. For a given k ̸ = 0, (1.1), known as Mordell’s equation, is an elliptic curve and has only finitely many solutions in integers by Siegel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of |k | was found early by M. Hall [2, 7] by means of a parametric family of the form (1.2) f(t) = t 9 (t9 +6t 6 +15t 3 + 12), g(t) = t15 27 + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27

### On primitive roots of 1 mod p k, divisors of p 2 − 1, Wieferich primes, and quadratic analysis mod p 3

, 2001

"... Primitive roots of 1 mod p k (k> 2 and odd prime p) are sought, in cyclic units group Gk ≡ AkBk mod p k, coprime to p, of order (p − 1)p k−1. ’Core ’ subgroup Ak has order p − 1 independent of precision k, and ’extension ’ subgroup Bk of all p k−1 residues 1 mod p is generated by p + 1. Integer divi ..."

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Primitive roots of 1 mod p k (k> 2 and odd prime p) are sought, in cyclic units group Gk ≡ AkBk mod p k, coprime to p, of order (p − 1)p k−1. ’Core ’ subgroup Ak has order p − 1 independent of precision k, and ’extension ’ subgroup Bk of all p k−1 residues 1 mod p is generated by p + 1. Integer divisors of powerful generators p ± 1 of ±Bk mod p k are investigated, as primitive root candidates. Fermat’s Small Theorem (FST): all n < p have n p ≡ n mod p is extended, or rather complemented, to: all proper divisors r | p 2 − 1 have r p ≡ / r mod p 3, a necessary (although not sufficient) condition for a primitive root mod p k>2. Hence 2 p ≡ / 2 mod p 3 for primes p> 2 (re: Wieferich primes [3] and FLT case1), and 3 p ≡ / 3 mod p 3 for p> 3. It is conjectured that at least one divisor of p 2 − 1 is a semi primitive root of 1 mod p k (k ≥ 3).

### On primitive roots of 1 mod p k, divisors of p ± 1, Wieferich primes, and quadratic analysis mod p 3

, 2001

"... Primitive roots of 1 mod pk (k> 2 and odd prime p) are sought, in cyclic units group Gk ≡ AkBk mod pk, coprime to p, of order (p − 1)pk−1. ’Core ’ subgroup Ak has order p − 1 independent of precision k, and ’extension ’ subgroup Bk of all pk−1 residues 1 mod p is generated by p + 1. Integer divisors ..."

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Primitive roots of 1 mod pk (k> 2 and odd prime p) are sought, in cyclic units group Gk ≡ AkBk mod pk, coprime to p, of order (p − 1)pk−1. ’Core ’ subgroup Ak has order p − 1 independent of precision k, and ’extension ’ subgroup Bk of all pk−1 residues 1 mod p is generated by p + 1. Integer divisors r, s of powerful generator p − 1 = rs of ±Bk mod pk, and of p + 1, are investigated as primitive root candidates. Using (p − 1) p−1 ≡ p + 1 mod p3, Fermat’s Small Theorem (FST) : x p−1 ≡ 1 mod p for 0 < x < p is extended and complemented to: divisors r | p ± 1 have distinct r p−1 mod p 3, so r p ≡ / r mod p 3 for each proper divisor, a necessary (but not sufficient) condition for a primitive root mod p k>2. Hence 2 p ≡ / 2 mod p 3 for prime p> 2 (re: Wieferich primes [3] and FLT case1) and 3 p ≡ / 3 mod p 3 for prime p> 3. It is conjectured that at least one divisor of p ± 1 is a semi primitive root of 1 mod p k.

### On primitive roots of unity, divisors of p 2 − 1, and an extension to mod p 3 of Fermat’s Small Theorem

, 2001

"... Primitive roots of 1 mod p k (k> 2 and odd prime p) are sought, in cyclic units group Gk ≡ AkBk mod p k, coprime to p, of order (p − 1)p k−1. ’Core ’ subgroup Ak has order p − 1 independent of precision k, and ’extension ’ subgroup Bk of all p k−1 residues 1 mod p is generated by p+1. Integer diviso ..."

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Primitive roots of 1 mod p k (k> 2 and odd prime p) are sought, in cyclic units group Gk ≡ AkBk mod p k, coprime to p, of order (p − 1)p k−1. ’Core ’ subgroup Ak has order p − 1 independent of precision k, and ’extension ’ subgroup Bk of all p k−1 residues 1 mod p is generated by p+1. Integer divisors of powerful generators p±1 of ±Bk mod p k are investigated, as primitive root candidates. Fermat’s Small Theorem (FST): all n < p have n p ≡ n mod p is extended, or rather complemented, to: all proper divisors r | p 2 − 1 have r p ≡ / r mod p 3, a necessary (although not sufficient) condition for a primitive root mod p k>2. Hence 2 p ≡ / 2 mod p 3 for primes p> 2 (re: Wieferich primes [3] and FLT case1), and 3 p ≡ / 3 mod p 3 for p> 3. It is conjectured that at least one divisor of p 2 − 1 is a semi primitive root of 1 mod p k (k ≥ 3). 1