On the day that one becomes an octogenarian, nature bestows a mathematical gift: a gradual reprieve from the relentlessly increasing likelihood that he or she will die in the coming year.

That gift may come as small comfort against the growing creakiness of joints and the still-mounting probability that the end is nigh. But an analysis of nearly 4,000 very long-lived Italians suggests that the rise in the risk of imminent death continues to slow until age 105. After that, researchers estimate, the chance of making it to see another birthday holds at about 50-50.

Perhaps it’s “nothing to blow a trumpet about,” said University of California, Berkeley demographer Kenneth Wachter. But at least the mortality rate levels off, the data suggest. Wachter and colleagues from universities in Italy and Germany published their findings in the journal Science.

For humankind in general, these findings hint at an intriguing, if largely theoretical, prospect: that the maximum possible human life span — essentially, the species’ design limit — has not yet been reached. It may even be extended by means as yet undiscovered.

If the “oldest old” tell us how long we could live, then many centenarians could, in principle, get even older. And maybe older still with the right elixir. “This data suggest our genetic heritage is permissive,” Wachter said. “Our bodies are not put together so that at some point, everything goes wrong.”

Indeed, he said, there’s reason to believe that some humans could beat the current longevity record of 122, which was set in 1997 by Jeanne Louise Calment of France.

The new research contributes to a debate that has preoccupied poets and philosophers for as long as they have set pen to paper. For instance, the historian Gaius Plinius Secundus, better known as Pliny the Elder, maintained a running tally of long-lived persons and pondered the significance of their longevity.

Scientists have been in the fray since at least 1825. That’s when the British actuary Benjamin Gompertz published the first models of human mortality and asked when, and whether, we must die.

In the new study, the international team of demographers and statisticians took advantage of the proliferation of people who live well past their 100th birthday. By calculating and analyzing the death rates of 3,836 cases of Italians who lived to 105 or beyond and combining them with existing data on mortality rates, the researchers created a model that reveals the statistical likelihood of death in every year of the human life span from 65 to 105.

Established demographic data show that after 65, people grow more likely to die with every year. And the math is unforgiving: Each year after 65, the probability of death rises at a pace that’s double what it was the previous year.

But when the researchers added the long-lived Italians to the earlier data, they saw that this doubling held up only until the average human’s 80th birthday. After that, the rate of increase began to slow. For the people who made it to 105, that annual increase in the probability of death seemed to stop.

The authors also showed that the annual mortality rate in those older than 105 declined slightly with each successive birth year, such that those born more recently tended to live longer.

This pattern “strongly suggests that longevity is continuing to increase over time and that a limit, if any, has not been reached,” wrote the team, which included demographer James W. Vaupel of the Max Planck Institute for Demographic Research in Germany.

“Our results contribute to a recently rekindled debate about the existence of a fixed maximum life span for humans,” they said. In any event, it raises “doubt that any limit is as yet in view.”

But University of Illinois epidemiologist S. Jay Olshansky, disagreed, saying, “The conclusion that they’ve come to, which is that there is no upper limit to life, is unreasonable.”

By the time people reach these extreme ages, at least half disappear every year. And since there are so few of them to begin with, this harsh reality “tells us the real story,” he said.

“If 100 people survive to age 110 out of billions — which is exactly what has happened — what difference does it make if it’s 50 or 60 that die before their next birthday?” he said.

Geneticist Jan Vijg of the Albert Einstein College of Medicine in New York believes he has seen the limit of the human life span, and said it ceased its upward trajectory in the 1990s with the death of Jean Calment. The Frenchwoman has the longest confirmed human life span at 122 years. “There is a ceiling. At the end of the day there is a ceiling,” Vijg said.

Vijg says he is “amazed” at the vigor of the scientific debate around an issue that is so distant from the reality of ordinary mortals. Improving the average life span of all humans — by extending gains in nutrition, creating new medicines and addressing the causes of infectious diseases — is a better way to spend one’s energy, he said. “We can improve quality of life more and maybe give more people more life.”