The key factor in predictions that Medicare will remain solvent until at least 2024 (at which point it will still be 87% solvent) is that increased longevity will result in more chronic disease, leading to significant increased costs.

For example,
"Alzheimer’s disease (AD)

The number of new patients diagnosed with Alzheimer’s disease is increasing, but Alzheimer’s-related mortality is decreasing. Together, these trends account for the predicted increase in the number of people living with Alzheimer’s from 5 million today to 16 million by 2050. This growth will profoundly impact Medicare costs, given that the average annual cost of a Medicare patient with Alzheimer’s is triple that of a patient without: $13,207 and $4,454, respectively.

In 2005, Medicare spent $91 billion on patients diagnosed with Alzheimer’s disease, and this amount is expected to more than double to $189 billion in 2015, and increase to over $1 trillion by 2050."

from http://healthcarecostmonitor.thehastingscenter.org/kimberlyswartz/projected-costs-of-chronic-diseases/
The contribution of other chronic diseases, stroke, diabetes, end stage renal disease, chronic lung disease and heart disease is discussed there.

The reason is that AD costs are increasing is that people are living longer which in turn increases their chances of AD.
For example, http://www.alz.org/alzheimers_disease_causes_risk_factors.asp
"The greatest known risk factor for Alzheimer’s is advancing age. Most individuals with the disease are age 65 or older. The likelihood of developing Alzheimer’s doubles about every five years after age 65. After age 85, the risk reaches nearly 50 percent."

The good news is that because our knowledge of the molecular mechanisms underlying diseases continues to improve (for example, see http://www.ncbi.nlm.nih.gov/pubmed/22323134), the chances to prevent such diseases increases as well and there is a good chance in 2024 we will not only have therapies for AD , but preventive measures as well. Thus, Medicare costs will not increase as projected and Medicare will not need to be "saved."

Of course, this is contingent on Congress continuing to support basic and translational research on these diseases, which is not certain given the GOP's desire to shrink the federal government's role to that it played in 1920, or perhaps even 1890.