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A300568 Numbers z such that z^8 = x^6 + y^7 for some integers x, y >= 1. 2
47775744, 22143375000, 23328000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also in the sequence: 4001504141376, 4275897935643, 11284439629824, 100192997081088, ... (with b = 63, 51, 72, 96, ... cf. below).

Consider a solution (x,y,z) of x^6 + y^7 = z^8. For any m, (x*m^28, y*m^24, z*m^21) is also a solution. Reciprocally, if (x/m^28, y/m^24, z/m^21) is a triple of integers for some m, then this is also a solution. We call primitive a solution for which there is no such m > 1.

When S = a^6 + b^14/4 is a square, then z = b^7/2 + sqrt(S) is a solution, with x = a*z and y = b*z. All known solutions are of this form.

Sequence A303268 lists the y-values, in the above case equal to b*z where z = a(n), with corresponding x = a*z = (a(n)^8 - A303268(n)^7)^(1/6).

LINKS

Table of n, a(n) for n=1..3.

EXAMPLE

a(1) = 47775744 = 2^16*3^6 is in the sequence because for this (2^16*3^6)^8 =: z^8 = (12*z)^7 + (288*z)^6 = (2^18*3^7)^7 + (2^21*3^8)^6.

a(2) = 22143375000 and a(3) = 23328000000 are in the sequence because for these z, we have z^8 = (30*z)^7 + x^6 with x = 1350*z resp. x = 1800*z.

PROG

(PARI) is(z)=for(y=1, sqrtnint(-1+z=z^8, 7), ispower(z-y^7, 6)&&return(y)) \\ returns the y-value (cf. A303268) if z is in this sequence, 0 else.

/* Code below for illustration only, not guaranteed to give a complete list. */

S=[]; N=1e5; forstep(b=1, 99, 1/2, forstep(a=1, N, 1/2, issquare(b^14/4+a^6, &r)&& !frac(z=b^7/2+r)&& !print("\n*** "z)&& S=setunion(S, [z])); print1([b])); S

CROSSREFS

Cf. A300564 (z^4 = x^2 + y^3) and A242183, A300565 (z^5 = x^3 + y^4), A300566 (z^6 = x^4 + y^5), A300567 (z^7 = x^5 + y^6), A302174.

See A303268 for the y-values.

Cf. A303376 (numbers of the form a^6 + b^7).

Sequence in context: A230756 A022238 A121491 * A210334 A022259 A290501

Adjacent sequences:  A300565 A300566 A300567 * A300569 A300570 A300571

KEYWORD

nonn,more,bref,hard

AUTHOR

M. F. Hasler, May 04 2018

STATUS

approved

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Last modified September 17 17:29 EDT 2021. Contains 347489 sequences. (Running on oeis4.)