Gastroesophageal Reflux Disease (GERD): This condition is characterized by stomach acid frequently flowing back into the tube connecting the mouth and stomach (esophagus). The symptoms of GERD, such as chest pain that radiates to the back and belching, align closely with the patient's complaints. The chronic nature of the symptoms over one month also supports this diagnosis, as GERD can cause persistent discomfort.

Peptic Ulcer Disease: This condition involves sores in the lining of the stomach or the first part of the small intestine. Peptic ulcers can cause chest pain and discomfort that may radiate to the back, along with symptoms like belching. The pain can be chronic and may worsen over time if not treated.
Functional Dyspepsia: Also known as non-ulcer stomach pain or indigestion, this condition involves recurring signs and symptoms of indigestion that have no obvious cause. Symptoms can include pain or discomfort in the upper abdomen or chest that may radiate to the back, along with belching.

Myocardial Infarction (Heart Attack): Although less likely given the description of the pain and the presence of belching, it's crucial not to miss a heart attack. Chest pain that radiates to the back can be a symptom of a heart attack, especially in women. Immediate medical attention is necessary if a heart attack is suspected.
Aortic Dissection: This is a serious condition where there is a tear in the inner layer of the aorta. It can cause severe, tearing chest pain that radiates to the back. Given the potential for high mortality, it's essential to consider this diagnosis, even though it's less common.
Pulmonary Embolism: A blockage in one of the arteries in the lungs can cause sudden and severe chest pain that may radiate to the back, among other symptoms. This condition requires immediate medical attention.

Esophageal Spasm: This condition involves abnormal muscle contractions of the esophagus, which can cause chest pain that may feel like it's radiating to the back. While it's less common, it could be considered if other diagnoses are ruled out.
Pancreatitis: Inflammation of the pancreas can cause severe abdominal pain that may radiate to the back. Although the primary symptom is abdominal pain, some patients may experience chest discomfort. This diagnosis would be considered rare in the context of the provided symptoms but should not be overlooked if other signs of pancreatitis are present.