A Quintic Problem (Posted on 2006-03-15)

	Submitted by K Sengupta
Rating: 2.3333 (3 votes)
Solution:	(Hide)
x=16 and y = +/- 16 are the only possible integer solutions to the equation under reference. EXPLANATION: From the given equation, we obtain: 5*(y^4) + 2560*(y^2) + 65536 = x^5 or, 32(5*(y^4) + 2560*(y^2) + 65536) = P^5, where P=2x or,(y+16)^5 = (y-16)^5 + P^5 .............(#) Now, we know that in accordance with Fermat's Last Theorem, the Diophantine equation : a^n + b^n = c^n where a, b, c and n are integers, has no nonzero solutions for n> 2. (Reference:http://mathworld.wolfram.com/FermatsLastTheorem.html). Hence, for integers ( positive, negative or zero) a, b and c; the equation a^5 + b^5 = c^5 is satisfied only in the following cases: Case (i) a=0 giving b=c, Case (ii) b=0 giving a=c Case (iii) c=0 giving a=-b Case (iv) a=b=c=0 In terms of equation (#), we can substitute, without any loss of generality: a = y-16, b= P and c = y+16. Now, in terms of case (i), y-16=0 and 2x = y + 16, so that y = 16 giving 2x = 32, so that x = 16. Accordingly, (x, y) = (16,16). In terms of case (ii), P =0 , or 2x = 0, giving x = 0, which is not feasible as x must be a non-zero integer ( whether positive or negative ), by conditions of the problem. This is a contradiction. In terms of case (iii), y+16 =0 and (y-16) = - P giving, y = -16 and (y - 16) = - 2x, so that: y = - 16 and x = -32/ -2 = 16. In terms of case (iv), P =0 giving x=0, which is not feasible for reasons explained earlier. Combining all the four cases we obtain x = 16 and y = +/- 16, which constitute the only possible solutions to the equation under reference. ------------Q E D -------------- Note: This problem is inspired by a previous puzzle submitted by Nick Hobson.

	Subject	Author	Date
	re: Small steps (another step)	Jer	2006-03-16 12:53:05
	graphical observation	Larry	2006-03-15 23:59:42
	Small steps	Gamer	2006-03-15 23:41:04
	Re:	Lithue	2006-03-15 23:29:34
	Predictions	tomarken	2006-03-15 18:23:46
	re: One solution, no proof	Vernon Lewis	2006-03-15 18:02:31
	One solution, no proof	tomarken	2006-03-15 15:15:43