This is related to the fact that the symmetric group 
  
    
      
        
          
            S
          
          
            n
          
        
      
    
    {\displaystyle \mathrm {S} _{n}}
   is a solvable group for n ⩽ 4 and not solvable for n ⩾ 5.
A lack of 1,25 dihydroxyvitamin D3 can result from a deficient dietary intake of vitamin D, or from a lack of exposure of the skin to sunlight, so the body cannot make its own vitamin D from cholesterol. The resulting hypovitaminosis D is usually due to a partial combination of both factors. Vitamin D3 (or cholecalciferol) is converted to 25-hydroxyvitamin D (or calcidiol) by the liver, from where it is transported via the circulation to the kidneys, and it is converted into the active hormone, 1,25 dihydroxyvitamin D3. Thus, a third cause of secondary hyperparathyroidism is chronic kidney disease. Here the ability to manufacture 1,25 dihydroxyvitamin D3 is compromised, resulting in hypocalcemia. Diagnosis
The gold standard of diagnosis is the PTH immunoassay. Once an elevated PTH has been confirmed, the goal of diagnosis is to determine whether the hyperparathyroidism is primary or secondary in origin by obtaining a serum calcium level:

Tertiary hyperparathyroidism has a high PTH and high serum calcium. It is differentiated from primary hyperparathyroidism by a history of chronic kidney failure and secondary hyperparathyroidism.Hyperparathyroidism can cause hyperchloremia and increase renal bicarbonate loss, which may result in a normal anion gap metabolic acidosis. Differential diagnosis
Familial benign hypocalciuric hypercalcaemia can present with similar lab changes. In this condition, the calcium creatinine clearance ratio, however, is typically under 0.01. Blood tests
Intact PTH
In primary hyperparathyroidism, parathyroid hormone (PTH) levels are either elevated or "inappropriately normal" in the presence of elevated calcium.