This is an intriguing finding, and I have no suggestions for either possible underlying mechanisms or possibilities for systematic error in the research. So far as I am aware the common risk factors for cancer are also risk factors for Alzheimer's disease (AD), so one would not expect to see the correlations shown here:
This was a cohort study in Northern Italy on more than 1 million residents. Cancer incidence was derived from the local health authority (ASL-Mi1) tumor registry and AD dementia incidence from registries of drug prescriptions, hospitalizations, and payment exemptions. Expected cases of AD dementia were calculated by applying the age-, sex-, and calendar year-specific incidence rates observed in the whole population to the subgroup constituted of persons with newly diagnosed cancers during the observation period (2004-2009). The same calculations were carried out for cancers in patients with AD dementia. Separate analyses were carried out for the time period preceding or following the index diagnosis for survivors and nonsurvivors until the end of 2009 and for different types and sites of cancer.
The risk of cancer in patients with AD dementia was halved, and the risk of AD dementia in patients with cancer was 35% reduced. This relationship was observed in almost all subgroup analyses, suggesting that some anticipated potential confounding factors did not significantly influence the results. The occurrence of both cancer and AD dementia increases exponentially with age, but with an inverse relationship; older persons with cancer have a reduced risk of AD dementia and vice versa. As AD dementia and cancer are negative hallmarks of aging and senescence, we suggest that AD dementia, cancer, and senescence could be manifestations of a unique phenomenon related to human aging.