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Let b be an odd number. By using elementary methods, we prove that: (1) When x is an odd number and y is an even number, the Diophantine equation (2x−1)(by−1)=2z2 has no positive integer solution except when b is two special types of odd number. (2) When x is an odd number and b≡±3(mod8), the Diophantine equation (2x−1)(by−1)=2z2 has no positive integer solution except where b=3 and is another special type of the odd number.

Mushrooms belong to the family “Fungi” and became famous for their medicinal properties and easy accessibility all over the world. Because of its pharmaceutical properties, including anti-diabetic, anti-inflammatory, anti-cancer, and antioxidant properties, it became a hot topic among scientists. However, depending on species and varieties, most of the medicinal properties became indistinct. With this interest, an attempt has been made to scrutinize the role of edible mushrooms (EM) in diabetes mellitus treatment. A systematic contemporary literature review has been carried out from all records such as Science Direct, PubMed, Embase, and Google Scholar with an aim to represents the work has performed on mushrooms focuses on diabetes, insulin resistance (IR), and preventive mechanism of IR, using different kinds of mushroom extracts. The final review represents that EM plays an important role in anticipation of insulin resistance with the help of active compounds, i.e., polysaccharide, vitamin D, and signifies α-glucosidase or α-amylase preventive activities. Although most of the mechanism is not clear yet, many varieties of mushrooms’ medicinal properties have not been studied properly. So, in the future, further investigation is needed on edible medicinal mushrooms to overcome the research gap to use its clinical potential to prevent non-communicable diseases.