Show that there are innitely many triples (x, y, z) of integers such t

1.	Show that there are innitely many triples (x, y, z) of integers such that x3 + y4 = z31.
Answer» Choose x = 2^4rand y = 2^3r. Then the left side is 2^12r+1. If we take z = 2^k, then we get 2^12r+1 = 2^31k. Thus it is suffcient to prove that the equation 12r + 1 = 31k has innitely many solutions in integers. Observe that (12 x 18) + 1 = 31 x 7. If we choose r = 31l + 18 and k = 12l + 7, we get 12(31l + 18) + 1 = 31(12l + 7); for all l. Choosing l ∈ N, we get innitely many r = 31l + 18 and k = 12l + 7 such that 12r + 1 = 31k. Going back we have innitely many (x, y, z) of integers satisfying the given equation.

Show that there are innitely many triples (x, y, z) of integers such that x3 + y4 = z31.

Answer»

Choose x = 2^4rand y = 2^3r. Then the left side is 2^12r+1. If we take z = 2^k, then we get 2^12r+1 = 2^31k. Thus it is suffcient to prove that the equation 12r + 1 = 31k has innitely many solutions in integers. Observe that (12 x 18) + 1 = 31 x 7. If we choose r = 31l + 18 and k = 12l + 7, we get

12(31l + 18) + 1 = 31(12l + 7);

for all l. Choosing l ∈ N, we get innitely many r = 31l + 18 and k = 12l + 7 such that 12r + 1 = 31k. Going back we have innitely many (x, y, z) of integers satisfying the given equation.

Discussion

No Comment Found

Related InterviewSolutions

What is the difference between multimolecular and macromolecular collids ? Give one example of each . How are associated colloids different from these two types of colloids ?
Gastrointestinal irritation, cardiovascular effects, including tachycardia,arrhythmias, and orthostatic hypotension, mental disturbances, and withdrawalare possible adverse effects of: A. Amantadine B. Benztropine C. Levodopa D. selegiline
Match the column I and IIColumn-IColumn-II(a)Parasitism(i)++(b)Commensalism(ii)+ –(c)Amensalism(iii)0 + (d)Mutualism(iv)0 –(1) (a) - iii, (b) - ii, (c) - iv, (d) - i (2) (a) - i, (b) - ii, (c) - iii, (d) - iv (3) (a) - ii, (b) - iii, (c) - i, (d) - iv (4) (a) - ii, (b) - iii, (c) - iv, (d) - i
Assertion: In commensalism, one organism is benefitted and other is unaffected Reason: Cattle egret bird and cattle is an example of commensalism (1) If both assertion and reason are true and reason is the correct explanation of assertion. (2) If both assertion and reason are true but reason is not the correct explanation of assertion. (3) If assertion is true but reason is false. (4) If both assertion and reason are false.
In charge of Calcutta Mint between 1832 to 1838, he studied and deciphered Ashokan rocks and pillars. His essays on antiquities are considered as classics. Hint: Varanasi. Who are we referring to?
Nature of climax community depends upon(A) Climate (B) Water (C) Soil fertility (D) Temperature
How do biofertilisers enrich the fertility of the soil?
Read carefully the passage and select the correct answer out of given four options:Biofertilizers are organisms that enrich nutrient quality of the soil. The important sources of biofertilizers are bacteria, fungi and cyanobacteria. The symbiotic association of Rhizobium with the root nodules of leguminous plants is the common example. These bacteria convert atmospheric nitrogen into nitrogenous compound and fix in the soil.The organisms which enrich the nutrient quality of the soil are called(a) Microbes(b) Biofertilizers(c) Yeast(d) Biopesticides
Electrophoresis is an essential procedure that needs to be employed in genetic fingerprinting why?
Assertion: Angiosperms form endosperm after fertilization and is diploid. Reason: In gymnosperms, endosperm is formed before fertilization and is triploid. a. Both assertion and reason are true, and reason is the correct explanation of assertion. b. Both assertion and reason are true, but reason is not the correct explanation of assertion. c. Assertion is true but reason is false. d. Both assertion and reason are false.