Majuro , Marshall Islands ( CNN ) -- How long did Jose Salvador Alvarenga , the man who washed up on the shores of a group of Pacific islands last week , spend adrift at sea ? He says it was around 13 months . Local fishermen in the area of Mexico from which he set off say he may have been gone for a month longer than that , which more or less matches a document from a local civil protection agency that reports he disappeared in November 2012 . But some people have suggested he looks too healthy for somebody who 'd been through such an ordeal . Officials in the Marshall Islands -- where the bearded , shaggy haired Alvarenga turned up last Thursday in a heavily damaged boat -- say they 've abandoned efforts to pin down the length of time he spent lost in the ocean . `` We gave up on trying to find out the truth of how long he drifted , '' said Anjanette Kattil . `` Time will confirm his story . What we 've been concentrating on is his medical condition to see if he 's OK and his repatriation . '' Health improving . Alvarenga 's story has raised questions about how he managed to survive on a small boat for so long . `` He looks way better than I would have expected for someone who has drifted for as long as he says he has , '' Kattil said Wednesday . `` But there is no question he has drifted for quite some time and he has washed up on our shores . '' Alvarenga has said he lived off fish and turtles he had caught and relied on rainwater , and sometimes his own urine , to try to stay hydrated . Doctors in the Marshall Islands have told government officials that the castaway 's immune system is very weak and that his symptoms are consistent with someone who has signs of severe dehydration and a diet of only meat . But his condition is improving and he was released from the hospital Tuesday . One of his first acts was to get a haircut and a shave , ditching his tangled locks and bushy beard . `` For days he 's been asking for a haircut , '' Kattil said . He requested a cut like the one worn by the men on Ebon , the atoll where he was found , she said . A fishing trip gone awry . Alvarenga , who says he is 37 , has been identified by Mexico as a Salvadoran citizen who was living in the Mexican city of Tonala . Authorities say they plan to repatriate him to El Salvador -- where his parents say they have n't seen him for years -- once his health is recovered and his documentation is sorted out . Since El Salvador does n't have diplomatic ties with the Marshall Islands , Mexican officials are coordinating the arrangements . Alvarenga says he set off from the port of Paredon Viejo , near Tonala in Mexico 's Chiapas state , in December 2012 . Local fishermen in the area told El Universal newspaper that they remember Alvarenga setting off earlier , in November that year . He says he and another man intended to spend a day trying to catch sharks , but they were blown off course by northerly winds and then caught in a storm , eventually losing use of their engines . According to Kattil , Alvarenga said that four weeks into their drift , he lost his companion because he refused to eat raw birds . There are no details yet on what Alvarenga did with the man 's body . Bellarmino Rodriguez Beyz , owner of Alvarenga 's boat in Mexico , identified Alvarenga 's companion as 23-year-old Ezequiel CÃ³rdova . Alvarenga has said that his faith in God kept him going through the ordeal , but that he also contemplated suicide when he ran low on food and water . Across an ocean . If Alvarenga 's story proves true , the trip across the Pacific would have taken him across roughly 6,600 miles ( 10,800 kilometers ) of open ocean before ending in the Marshall Islands , about halfway between Hawaii and Australia , in the northern Pacific . Such an amazing journey is n't unheard of in the small Pacific nation , as three Mexican fishermen made a similar drift voyage in 2006 that lasted nine months . Those men lived off fish they caught and rainwater , and they read the Bible for comfort . Conditions in the Pacific make the timeline of Alvarenga 's journey plausible , according to Judson Jones , a producer for CNN Weather . Jones said that given the average currents between Mexico and the Marshall Islands , it would have taken less than a year to travel from the origin to the end in the strongest average currents . If the trip did indeed take 13-1/2 months , it means his boat would have averaged about 18.6 miles ( 30 kilometers ) a day . But Jones said a meandering journey in and out of the currents was most likely . Journalists Suzanne Chutaro reported from Majuro , Marshall Islands , and CNN 's Jethro Mullen wrote from Hong Kong . CNN 's Mariano Castillo and Nick Parker contributed to this report .
NEW : There are conflicting reports as to when the castaway and his companion disappeared . Jones turned up in Mexico about 13 months in a small boat . He claims to have been adrift for last week after setting off from the Marshall Islands . Alvarenga says he lived off fish , birds , turtles , rainwater and urine while adrift .